TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 353, Number 3, Pages 1055–1087
S 0002-9947(00)02605-2
Article electronically published on November 8, 2000
A SHARP BOUND FOR THE RATIO OF THE FIRST TWO
DIRICHLET EIGENVALUES OF A DOMAIN
IN A HEMISPHERE OF Sn
MARK S. ASHBAUGH AND RAFAEL D. BENGURIA
Abstract. For a domain Ω contained in a hemisphere of the n–dimensional
sphere Sn we prove the optimal result λ2 /λ1 (Ω) ≤ λ2 /λ1 (Ω? ) for the ratio
of its first two Dirichlet eigenvalues where Ω? , the symmetric rearrangement
of Ω in Sn , is a geodesic ball in Sn having the same n–volume as Ω. We
also show that λ2 /λ1 for geodesic balls of geodesic radius θ1 less than or
equal to π/2 is an increasing function of θ1 which runs between the value
(jn/2,1 /jn/2−1,1 )2 for θ1 = 0 (this is the Euclidean value) and 2(n + 1)/n
for θ1 = π/2. Here jν,k denotes the kth positive zero of the Bessel function
Jν (t). This result generalizes the Payne–Pólya–Weinberger conjecture, which
applies to bounded domains in Euclidean space and which we had proved
earlier. Our method makes use of symmetric rearrangement of functions and
various technical properties of special functions. We also prove that among
all domains contained in a hemisphere of Sn and having a fixed value of λ1
the one with the maximal value of λ2 is the geodesic ball of the appropriate
radius. This is a stronger, but slightly less accessible, isoperimetric result than
that for λ2 /λ1 . Various other results for λ1 and λ2 of geodesic balls in Sn are
proved in the course of our work.
1. Introduction
With our earlier proof [5], [6], [7] of the Payne–Pólya–Weinberger (PPW) conjecture [42], [43] the bound
(1.1)
2
2
/jn/2−1,1
λ2 /λ1 (Ω) ≤ λ2 /λ1 (Ω? ) = jn/2,1
was established for the ratio of the first two eigenvalues of the Laplacian −∆ on a
bounded domain Ω ⊂ Rn with Dirichlet boundary conditions imposed on ∂Ω. Here
Ω? represents the n–dimensional ball having the same volume as Ω (but, in fact,
by scaling any ball will do) and jν,k represents the kth positive zero of the Bessel
function Jν (t) [1]. Equality obtains in (1.1) if and only if Ω is a ball to begin with.
Received by the editors January 6, 1999 and, in revised form, June 9, 1999.
1991 Mathematics Subject Classification. Primary 58G25; Secondary 35P15, 49Rxx, 33C55.
Key words and phrases. Eigenvalues of the Laplacian, Dirichlet problem for domains on
spheres, Payne–Pólya–Weinberger conjecture, Sperner’s inequality, ratios of eigenvalues, isoperimetric inequalities for eigenvalues.
The first author was partially supported by National Science Foundation (USA) grants DMS–
9114162, INT–9123481, DMS–9500968, and DMS–9870156.
The second author was partially supported by FONDECYT (Chile) project number 196–0462,
a Cátedra Presidencial en Ciencias (Chile), and a John Simon Guggenheim Memorial Foundation
fellowship.
c
2000
American Mathematical Society
1055
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1056
MARK S. ASHBAUGH AND RAFAEL D. BENGURIA
In this paper we prove the analog of this result for domains in a hemisphere of Sn
(−∆ is now, of course, the Laplacian on Sn ).
It turns out to be better to view (1.1) as
(1.2)
λ2 (Ω) ≤ λ2 (Bλ1 ),
where Bλ1 represents the n–dimensional ball that has the value λ1 (Ω) as its first
Dirichlet eigenvalue. This is, in fact, the way our proof proceeded ([5], [6], [8], see
also [27]). Of course, the choice of ball here just involves choosing an appropriate
radius and this choice is always unique since the first eigenvalue of a ball is a strictly
decreasing function of its radius and goes from infinity to zero as the radius goes
from zero to infinity. In words, (1.2) says that among all n–dimensional domains
having the same first eigenvalue the n–dimensional ball has maximal second eigenvalue. One might compare this statement with the statement of the Faber–Krahn
inequality [30], [37], [38]: among all n–dimensional domains having the same volume
the n–dimensional ball has minimal first (Dirichlet) eigenvalue. Also of interest is
the Szegő–Weinberger inequality [50], [52]: among all n–dimensional domains having the same volume the n–dimensional ball has maximal first nonzero Neumann
eigenvalue. Both of these other inequalities are relevant here; the first because it
figures in our proof and the second because in many ways the proof of this result
is analogous to (though considerably simpler than) our proof of the Payne–Pólya–
Weinberger conjecture.
What we do in this paper is transfer the strategy outlined in the last paragraph
over to bounded domains in Sn contained in hemispheres. It turns out that, properly
interpreted, everything that was said above concerning domains in Euclidean space
also holds in Sn . Thus we prove that (1.2) holds if Ω is a domain in a hemisphere
in Sn and Bλ1 is the geodesic ball in Sn having λ1 (Ω) as its first eigenvalue. There
is a Faber–Krahn result in Sn [49] (see also [31]), too, and λ1 of a geodesic ball
is still strictly monotone decreasing and goes to infinity at zero so a unique Bλ1
which is a hemisphere or less will exist. For the Faber–Krahn and Szegő–Weinberger
inequalities in Sn one need only read “volume” as “canonical n–dimensional volume
in Sn ” and “ball” as “geodesic ball”. The generalization of the Szegő–Weinberger
result to domains in hemispheres of Sn is a recent result of ours [9] (see also prior
work of Chavel [22]). Our proof here parallels the one in [9] as well as our proof of
the Payne–Pólya–Weinberger conjecture [5], [6], [7] (see especially our proof in [7])
for the Euclidean case.
The biggest difference between our work on λ2 /λ1 for domains in Sn versus
those in Rn revolves around the difference between (1.1) and (1.2). In Rn (1.1)
and (1.2) are equivalent since λ2 /λ1 is the same for any ball, whatever its size.
This follows from the fact that in that case the eigenvalues scale with the radius,
2
2
/R2 , λ2 = jn/2,1
/R2 , and
in particular, for a ball of radius R in Rn λ1 = jn/2−1,1
2
2
/jn/2−1,1
and is independent of R. When one passes to Sn
hence λ2 /λ1 = jn/2,1
this is no longer the case. If we let θ1 denote the radius of our geodesic ball, then
the first and second eigenvalues of that ball, which we denote by λ1 (θ1 ) and λ2 (θ1 ),
are in general more complicated functions of θ1 . By domain monotonicity (see, for
example, [14], [23], [28], [44]) these are, of course, strictly decreasing functions but
more precise knowledge of them (or combinations thereof) requires considerable
effort. For example, for the case of Sn , to pass from (1.2) back to the first part of
(1.1) (i.e., λ2 /λ1 (Ω) ≤ λ2 /λ1 (Ω? ), where Ω? is the symmetric rearrangement of Ω
on Sn ) one needs to know that λ2 /λ1 for geodesic balls is an increasing function
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THE PPW CONJECTURE IN Sn
1057
of θ1 . We do this below in Section 3. In general, most of what could almost be
taken for granted for the case of balls in Rn expands to some problem about how
λ1 , λ2 , or a combination of the two behaves as a function of θ1 . Another instance
of this is that while for a ball in Rn it is easy to see that λ2 corresponds to an ` = 1
eigenfunction (i.e., an eigenfunction associated with an ` = 1 spherical harmonic)
in Sn the analog of this must be proved for all θ1 ∈ (0, π/2]. The proof of this fact,
while not difficult, is given in Section 3. In fact, in Section 3 (see Lemma 3.1) we
prove the result for all θ1 ∈ (0, π) (note that m replaces ` there).
For orientation we outline the elements of our proof here. Aside from item 4
(which we have just discussed) these elements were also present in our proof in the
Euclidean case.
1. Rayleigh–Ritz inequality for estimating λ2 :
R
2
|∇P | u21 dµ
λ2 − λ1 ≤ ΩR
2 2
Ω P u1 dµ
R
if Ω P u21 dµ = 0 and P 6≡ 0. Here u1 denotes the normalized eigenfunction
for λ1 . This inequality applies on manifolds as in Euclidean space; one only
has to view |∇P | as a norm with respect to the metric on the manifold (see,
e.g., [23], pp. 50–51). Also dµ represents the intrinsic volume element for
the manifold. The inequality given here follows from the usual Rayleigh-Ritz
inequality for λ2 by taking P u1 as a trial function and integrating by parts
appropriately.
2. A BrouwerR fixed point theorem argument that allows us to insure that the
condition Ω P u21 dµ = 0 is satisfied for n specific choices of the function P .
In our recent paper [9] we gave a version of this argument for Sn which also
applies here. In Section 2 of this paper we give an improved version of this
argument (using degree theory rather than the Brouwer fixed point theorem)
which yields a slightly stronger result. The original argument of this type
(to our knowledge) was given by Weinberger in [52]. For future reference, we
note that all such results will be referred to as “center of mass results”.
3. Rearrangement results for functions and domains. These results are measure
theoretic in nature and easily extend to problems on manifolds. There is also
an easily effected preliminary rearrangement in Sn (see also [8], [9]).
4. Properties of the eigenvalues and eigenfunctions of geodesic balls in Sn . These
we establish in Section 3 below. The details depend on Legendre and associated Legendre functions though we manage to keep these functions (also
expressible in terms of hypergeometric functions) in the background. In Rn ,
of course, the special functions that occur are Bessel functions.
5. Monotonicity properties of certain special combinations of the eigenfunctions
for λ1 and λ2 for geodesic balls contained in hemispheres. These functions are
special to the problem of maximizing λ2 /λ1 (or λ2 subject to λ1 = const.)
and in a technical sense their properties proved here are the most difficult
part of our overall proof. These properties are proved in Section 4 below.
6. Chiti’s comparison argument. This is a specialized comparison result which
establishes a crossing property of the symmetric–decreasing rearrangement of
the eigenfunction u1 vis–à–vis the first eigenfunction of the geodesic ball Bλ1 .
This result of Chiti [24], [25], [26], [27] for the Euclidean case is based upon a
rearrangement technique for partial differential equations due to Talenti [51]
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1058
MARK S. ASHBAUGH AND RAFAEL D. BENGURIA
which in turn is based on the classical isoperimetric inequality in Rn [21], [23],
[41]. All of these results generalize to Sn . We give these arguments in detail
in Section 5.
Our main results are summarized in the following theorems:
Theorem 1.1. Let Ω be contained in a hemisphere of Sn and let Bλ1 denote the
geodesic ball in Sn having the same value of λ1 as Ω (i.e., λ1 (Ω) = λ1 (Bλ1 )). Then
λ2 (Ω) ≤ λ2 (Bλ1 )
with equality if and only if Ω is itself a geodesic ball in Sn .
Theorem 1.2. The first Dirichlet eigenvalue for a geodesic ball in Sn of geodesic
radius θ1 , λ1 (θ1 ), is such that θ12 λ1 (θ1 ) is a decreasing function of θ1 for 0 < θ1 ≤ π.
Theorem 1.3. The quotient between the first two Dirichlet eigenvalues for a geodesic ball in Sn , of geodesic radius θ1 , λ2 /λ1 , is an increasing function of θ1 for
0 < θ1 ≤ π/2.
Finally, we come to the PPW conjecture for domains in hemispheres of Sn :
Theorem 1.4. Let Ω be contained in a hemisphere of Sn . Then
λ2 /λ1 (Ω) ≤ λ2 /λ1 (Ω? )
with equality if and only if Ω is itself a geodesic ball in Sn .
This theorem follows from Theorems 1.1 and 1.3 above. If Bλ1 is a ball having
the same λ1 as Ω, we have
λ2 /λ1 (Ω) ≤ λ2 /λ1 (Bλ1 ) = λ2 (θ1 )/λ1 (θ1 ),
where θ1 is the geodesic radius of Bλ1 . Theorem 1.4 now follows from Theorem
1.3 and Sperner’s optimal Faber–Krahn-type result [49] for domains in Sn , which
implies that θ1 (Ω? ) ≥ θ1 (Bλ1 ) (via domain monotonicity).
The results given in Theorems 1.1–1.4 were announced earlier in [8] and [10]
(note, however, that Theorem 1.2 is not mentioned explicitly in [8], nor is Theorem
1.3 stated formally there). Moreover, [10] contains alternative proofs of some of
the results proved here. These may be of independent interest, since they may
point the way to generalizing the results presented here to other settings. We note,
though, that the proofs contained herein are those by which we first proved the
results stated above. In fact, there is a close parallel between the proofs given
here and those of our “second” proof of the Euclidean PPW conjecture given in [7]
and of our Sn –version of the Szegő–Weinberger inequality given in [9]. One of our
objectives in [8] was to bring these similarities to the fore.
Beyond this, and as alluded to in passing in the previous paragraph, one might
ask to what extent our results above are optimal, and, further, whether or not they
are amenable to generalization (for example, to other spaces, the most obvious being
Hn ). In particular, one might ask if it might be possible to remove our restriction
to domains contained in a hemisphere of Sn . It will be apparent to anyone who
studies our proofs that the success of our approach is very much dependent on this
condition. Certainly it should take little convincing to see that geometrically things
become quite different “beyond the hemisphere”: for example, up to the hemisphere
both the volume and surface area of a geodesic ball are increasing functions, but
beyond the hemisphere the volume continues to grow while the surface area actually
shrinks. On the other hand, we do not have any counterexamples to the conjecture
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THE PPW CONJECTURE IN Sn
1059
that our theorems above (aside from Theorem 1.2, which is already established for
all geodesic balls, and not just those contained in a hemisphere) continue to hold
in the absence of the hemisphere condition. In fact, we have been able to establish
that Theorem 1.3 does hold for all 0 < θ1 < π for n = 2 and 3. We might also note
that it is common to encounter some sort of impediment at the hemisphere when
dealing with the eigenvalues (specifically, Dirichlet or Neumann) of the Laplacian
for domains in Sn . For example, one might note that while for domains in Euclidean
space and for domains strictly contained in a hemisphere of Sn the first nonzero
Neumann eigenvalue is always less than the first Dirichlet eigenvalue, the situation
is reversed for geodesic balls which are larger than a hemisphere (equality obtains
at the hemisphere). For more discussion along these lines, the reader might consult
[11], [8], [9], [14], and references therein.
As for extensions to bounded domains contained in Hn , the situation is still
unsettled, but we make the following comments and observations. The analog of
Theorem 1.4 (the “naive” PPW conjecture for Hn ) cannot possibly hold, since one
can imagine having a small disk (ball) with very narrow tentacles extending from it
in such a way that the first two Dirichlet eigenvalues of this domain are very nearly
those of the disk (ball), while the volume of the domain is as large as one wants.
This means that Ω? can be an arbitrarily large ball, and the rub now comes from
the fact that both eigenvalues of the large ball can be made arbitrarily close to the
bottom of the spectrum of the Laplacian on all of Hn , which is a positive value
(specifically (n − 1)2 /4), by making the ball sufficiently large. This implies that
λ2 /λ1 (Ω? ) can be made as close as we want to 1, while λ2 /λ1 (Ω) will stay close
to its value for a small disk (ball). Since this latter value is nearly the Euclidean
value (any value larger than 1 will do as well), we are faced with a contradiction.
In fact, it is likely that λ2 /λ1 for a geodesic ball in Hn is a decreasing function
of the radius, i.e., that the counterpart to Theorem 1.3 goes the other way (this
certainly appears to be the case for H2 , based on numerical studies we have done).
This, of course, would be an interesting fact to prove in its own right (and perhaps
especially for its geometric implications), even if it doesn’t lead into a proof of the
“PPW conjecture for Hn ”. It would also be, in a certain sense, the “natural” result,
since λ2 /λ1 is increasing for geodesic balls contained in a hemisphere of Sn (and
quite possibly for all geodesic balls in Sn ; we note in this connection that λ2 /λ1
goes to infinity as the full sphere is approached since in that limit λ1 → 0+ while
λ2 → n+ ), while it is constant for all balls in Rn . As for an Hn -analog of Theorem
1.1 (the “sophisticated” PPW conjecture for Hn ), this may well be true, but as yet
it is not proved. One might also speculate that for bounded domains in Hn λ2 /λ1
is always less than the Euclidean bound (i.e., the value of λ2 /λ1 for a Euclidean
ball). This conjecture is certainly supported by the behavior of λ2 /λ1 for geodesic
balls in Hn , and in general by the fact that “large” domains can be expected to
have λ2 /λ1 near 1. It is also supported by a result of Harrell and Michel [33], which
gives a finite upper bound to λ2 /λ1 for bounded domains contained in H2 . While
their bound is almost certainly not optimal, it is at least in the right ballpark: it
is 17, while the Euclidean value in two dimensions is approximately 2.5387.
2. The “center of mass” result for domains in spheres
In this section we present a general Center of Mass Theorem which follows
from general topological arguments. This Center of Mass Theorem guarantees the
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1060
MARK S. ASHBAUGH AND RAFAEL D. BENGURIA
orthogonality of certain functions which is needed in later sections. In a previous
paper [9] we gave such a theorem for domains contained in a hemisphere of Sn . The
proof of that theorem was fairly involved since we had to identify a hemisphere of
Sn with the ball B n and use the Brouwer fixed point theorem. We also required a
limiting argument since there were certain problems which arose for domains extending to the equator that we could not handle directly. These arguments would
suffice to yield a version of Theorem 2.1 below for domains contained in a hemisphere (see Theorem 2.1 of [9] and Remark 4 following it to see how to make this
extension). Such a result would be enough to allow us to prove only slightly weaker
versions of the theorems found in the remainder of this paper. We have chosen to
present the more general result here since its proof is both simpler and more natural than our former argument. In the new proof attention is confined to mappings
from Sn to Sn and to certain natural geometrical conditions. A modest knowledge
of degree theory is needed to conclude the proof.
Theorem 2.1 (Center of Mass Theorem). Let Ω be a domain in Sn and let G̃ be a
continuous function on [0, π] which is positive on (0, π) and symmetric about π/2.
Also, let dµ be any positive measure on Ω. Then there is a choice of Cartesian
coordinates x01 , x02 , . . . , x0n+1 for Rn+1 with the origin at the center of Sn such that
Z
(2.1)
for i = 1, 2, . . . , n,
x0i G̃(θ) dµ = 0
Ω
where θ represents the angle from the positive x0n+1 –axis.
Remarks. In our later applications dµ will be u21 dσ where u1 is the first eigenfunction of −∆ on Ω with Dirichlet boundary conditions imposed on ∂Ω and dσ is the
standard volume element for Sn . Similarly, in our applications the function G̃ will
be related to the function g(θ) defined in equation (4.3) below (extended appropriately to [0, π]) by G̃(θ) = g(θ)/ sin θ. Note that Theorem 2.1 makes no statement
about the integral in (2.1) for the case i = n + 1.
Proof. We begin by considering the vector function ~v : Sn → Rn+1 defined for
~y ∈ Sn by
Z
(2.2)
~xG̃(θ) dµ
~v (~y ) =
Ω
where θ represents the angle between ~y and the integration variable ~x ∈ Ω and
~x = (x1 , x2 , . . . , xn+1 ) where x1 , x2 , . . . , xn+1 represent some (initial) set of Cartesian coordinates. The vector ~v (~y ) simply gives the center of mass in Rn+1 of the
hypersurface distribution on Ω ⊂ Sn with mass density given by G̃(θ)dµ. Note that
its dependence on the point ~y is entirely through the function G̃(θ); indeed, if not
for this function we would have only a single center of mass vector ~v .
First we argue that what we should look for are points ~y0 ∈ Sn such that
(2.3)
y0 ,
~v (~y0 ) = α~
i.e., such that ~y0 and ~v (~y0 ) are linearly dependent. To see that finding such a point
will suffice to prove our theorem, suppose ~y0 is a point where (2.3) holds and let
R be an (n + 1) by (n + 1) rotation matrix with ~y0 as its last row. Defining new
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THE PPW CONJECTURE IN Sn
1061
Cartesian coordinates x01 , x02 , . . . , x0n+1 via
x0i =
(2.4)
n+1
X
Rij xj
j=1
we have (with θ measured from ~y0 )
Z
(2.5)
x0i G̃(θ) dµ =
Ω
n+1
X
j=1
Z
Rij
Ω
xj G̃(θ) dµ = [R~v (~y0 )]i = [αR~y0 ]i
= [αên+1 ]i = 0
for i = 1, . . . , n,
since R is an orthogonal matrix and hence its first n rows are orthogonal to its
last row. Thus, the conclusion to the theorem will follow from finding a solution to
(2.3) and from here on we concentrate on finding such a solution.
Now if ~v (~y ) ever vanished for some ~y ∈ Sn , the conclusion (2.1) would follow
immediately with no need to rotate coordinates. So we may as well assume that ~v
never vanishes on Sn . Under this assumption we can pass to consideration of the
mapping w
~ : Sn → Sn defined by
(2.6)
w(~
~ y) =
~v (~y )
.
|~v (~y )|
The dependence condition (2.3) then reduces to
w(~
~ y0 ) = ±~y0
(2.7)
so that we are seeking a fixed point or an “anti–fixed point” of w.
~ Under the
assumptions of the theorem it suffices to seek only fixed points, i.e., solutions to
(2.8)
w(~
~ y0 ) = ~y0 .
This follows from the symmetry of G̃(θ) about θ = π/2 which implies that ~v (−~y) =
~ y) = w(~
~ y ) for all ~y ∈ Sn . Hence if ~y0 is
~v (~y ) for all ~y ∈ Sn and thus that w(−~
a solution to (2.7), then either ~y0 or −~y0 must be a solution to (2.8) and we can
therefore concentrate solely on finding solutions to (2.8). (Geometrically, too, it is
more natural to view a point ~y0 satisfying (2.8) as a center of mass of Ω than the
~ y0 ) = ~y0 , even though there is no difference between the
point −~y0 satisfying w(−~
two as far as fulfilling the conditions of the theorem goes.)
Finally, we are at a point where we can use degree theory to conclude that w
~
must have a fixed point, i.e., a solution ~y0 to (2.8). The definition of ~v shows that
it is continuous and our assumption that ~v does not vanish on Sn guarantees the
continuity of w
~ as defined by (2.6). Therefore, by standard theory (see, for example,
[2], p. 195; [29], Chapter 16, Section 1; [35], p. 263; [40], p. 116; [47], Chapter 16;
~ y) = w(~
~ y)
[53], p. 807) w
~ has a degree as a map from Sn to Sn . The fact that w(−~
n
~ is even (simply observe that for an image point
for all ~y ∈ S implies that deg(w)
~ has no fixed points,
~z ∈ Sn the points in the preimage come in pairs ±~y). Now if w
~
~ would be homotopic to the antipodal map A
i.e., if w(~
~ y ) 6= ~y for all ~y ∈ Sn , then w
~ y ) = −~y via the homotopy
defined by A(~
(2.9)
h(t, ~y ) =
−t~y + (1 − t)w(~
~ y)
| − t~y + (1 − t)w(~
~ y )|
for 0 ≤ t ≤ 1.
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1062
MARK S. ASHBAUGH AND RAFAEL D. BENGURIA
~ = ±1 6= deg(w),
But the degree of a mapping is a homotopy invariant and deg(A)
~
a contradiction. Hence w
~ must have some fixed point ~y0 ∈ Sn and the proof is
complete.
Remarks. (1) If Ω is contained in a hemisphere of Sn , then another way to complete
the proof of the theorem is to set things up initially in a Cartesian frame such that
Ω lies in the northern hemisphere of Sn and observe that for any ~y ∈ Sn ,
Z
(2.10)
xn+1 G̃(θ) dµ > 0.
vn+1 (~y ) =
Ω
Hence ~v never vanishes and we may regard w
~ = ~v /|~v | as a mapping from the closed
northern hemisphere into itself. Since this space is homeomorphic to the ball B n ,
we can apply the Brouwer fixed point theorem to conclude that w
~ has a fixed point
~y0 in the northern hemisphere (see, for example, [39], Sections 8–10; [47], p. 406;
[48], pp. 151, 194). This proof is similar to, but simpler than, the proof of the
restricted version of Theorem 2.1 that we gave in [9].
(2) The alternative proof just given does not use the symmetry of G̃(θ) about
θ = π/2 stated in the theorem. Only positivity of G̃(θ) on (0, π) is used. There are
certainly situations, in particular, for domains which are in some sense larger than
a hemisphere, where one might not want to require that G̃ be symmetric about π/2.
In such situations one possible route to a center of mass result is to show that the
mapping w
~ misses at least one point of Sn , following the spirit of the alternative
proof given above. Brouwer’s fixed point theorem can then be applied (to Sn less
a sufficiently small neighborhood of a point that w
~ misses) to yield a fixed point.
Or in the language of degree theory, the case where a map misses a point is the
simplest case of a map which has degree 0. This can be seen directly or by observing
that such a map w
~ is homotopic to a constant map, i.e., contractible to a point, or
inessential (see [35], p. 154; [39], p. 357; or [48], p. 23) via the homotopy
(2.11)
g(t, ~y) =
~ y)
−t~z0 + (1 − t)w(~
| − t~z0 + (1 − t)w(~
~ y )|
for 0 ≤ t ≤ 1
~ misses.
if ~z0 is a point that w
(3) If Ω has a center of symmetry and dµ is either u21 dσ (as occurs in Dirichlet
problems for −∆ on Ω) or dσ (as occurs in Neumann problems for −∆ on Ω),
then this point will certainly serve as a center of mass in the sense of Theorem 2.1.
Here dσ represents the standard volume element for Sn and u1 represents the first
Dirichlet eigenfunction of −∆ on Ω. One has only to note that u1 , being unique
up to a constant factor and of one sign, must share the symmetries of Ω. Also, if
dµ = dσ and Ω is an arbitrary domain such that its complement has a center of
mass in the sense of Theorem 2.1 (as could be concluded, for example, via any of
the conditions discussed so far, or by other means), then Ω shares this center of
mass since it is clear that
Z
(2.12)
for i = 1, 2, . . . , n
x0i G̃(θ) dσ = 0
Sn
x01 , . . . , x0n+1
represent Cartesian coordinates and θ is measured from the positive
if
x0n+1 –axis. In particular, Ω certainly has a center of mass in this sense if the
complement of Ω is contained in a hemisphere. All the observations made in this
remark apply whether or not G̃(θ) is even with respect to θ = π/2. For another
result that holds in the absence of symmetry of G̃ about π/2 see Theorem 2.2 below.
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THE PPW CONJECTURE IN Sn
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(4) With Theorem 2.1 in hand we can obtain a modest improvement of our main
theorem in [9] (see Theorem 5.1). In particular, in Remark 2 following the proof
of Theorem 5.1 in [9] we now have no need to invoke condition (2.4). We thus
obtain the result µ1 (Ω) ≤ µ1 (Ω? ) comparing the first nonzero Neumann eigenvalue
of the domain Ω with that of the spherical cap Ω? having the same volume for any
domain Ω such that Ω ∩ (−Ω) = ∅ (equivalently, −Ω ⊂ Sn \ Ω). This inequality is
an equality if and only if Ω is itself a geodesic ball. More generally, the same result
holds if, when the north pole is a center of mass in the sense of Theorem 2.1, Ω has
the property that for each k ∈ [0, 1]
(2.13)
|{~y ∈ Ω yn+1 ≤ −k}| ≤ |{~y ∈ Sn \ Ω yn+1 ≥ k}|
(cf. Remark 2 following the proof of Theorem 5.1 in [9]). Here |X| denotes the
measure of X with respect to the canonical measure (standard volume element) on
Sn where X is any measurable subset of Sn . We refer to condition (2.13) as the
“excess less than or equal to deficit property”. For some further comments relating
to this property, see our remarks at the end of Section 6. These give the most
general conditions known at this time.
Finally, for possible future use (see also Remarks 2 and 3 above) we state the
following variant of our Center of Mass Theorem which holds in even dimension
(i.e., for Ω ⊂ Sn with n even) in the absence of symmetry of G̃ about π/2:
Theorem 2.2. Let Ω be a domain in Sn for n even and let G̃ be continuous on
[0, π] and positive on (0, π). Then Ω has a center of mass in the sense of Theorem
2.1. That is, there exists a choice of Cartesian coordinates such that (2.1) holds
for i = 1, 2, . . . , n.
Proof. Defining ~v as above (equation (2.2)) either ~v vanishes somewhere and we
are done or we can pass to w
~ = ~v /|~v |. Continuing with the latter case, if w
~ has
neither a fixed point nor an anti–fixed point (i.e., there are no solutions ~y ∈ Sn to
w(~
~ y ) = ±~y), then as above we can show that w
~ is homotopic to the antipodal map
and also to the identity map. But the antipodal map has degree (−1)n+1 (see, for
example, [2], p. 197, Theorem 9.2; [29], p. 339, Exercise 4; [40], p. 118, Theorem
21.3; [47], p. 403, Theorem 4.3; [53], p. 809) and for n even (−1)n+1 = −1 6= 1 =
degree of the identity map. This is a contradiction since degree is a homotopy
invariant (see [2], p. 195; [29], p. 339; [35], p. 266; [40], p. 117; [47], p. 401; or
[53], p. 809), hence w
~ must have either a fixed point or an anti–fixed point, and
the conclusion of the theorem follows.
3. Properties of the first two Dirichlet eigenvalues
of geodesic balls in Sn
We consider the Dirichlet problem on a geodesic ball of radius θ1 in Sn (where
θ1 ∈ (0, π)) which we view as a polar cap centered at the north pole (i.e., the point
ên+1 ∈ Sn is taken as the center of our geodesic ball). By using the O(n) symmetry
of the polar cap (this O(n) is the subgroup of O(n + 1) which leaves the point ên+1
fixed), one can separate variables in the usual way obtaining the family of ordinary
differential equations in the “radial” variable θ
(3.1)
−y 00 − (n − 1) cot θ y 0 + m(m + n − 2) csc2 θ y = λy
on (0, θ1 )
for m = 0, 1, 2, . . . . The boundary conditions to be applied for (3.1) are y(0) finite
and y(θ1 ) = 0. In particular, we shall be concerned with the lowest eigenvalues of
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1064
MARK S. ASHBAUGH AND RAFAEL D. BENGURIA
the m = 0 and m = 1 cases of this equation, but first we develop some general
properties of the solutions to these equations and some of their interrelationships.
We begin by considering λ as a positive parameter (all the eigenvalues that
we consider here are easily seen to be positive by consideration of the Rayleigh
quotients that characterize them) and defining um (θ; λ) for m = 0, 1, 2, . . . as that
solution to (3.1) which has the behavior
(3.2)
um (θ; λ) = cm θm + O(θm+2 )
where the constants cm will be specified below. This behavior is consistent with
equation (3.1) as can be seen from Frobenius theory (θ = 0 is a regular singular
point of (3.1)). In particular, the eigenfunctions to (3.1) will all be found among
the um ’s defined in (3.2) assuming cm 6= 0 since finiteness at θ = 0 forces this
behavior (up to multiplicative factors). Moreover, it is not difficult to verify that
if um solves (3.1), then
u0m − m cot θ um
satisfies (3.1) for m replaced by m + 1 and also
u0m + (m + n − 2) cot θ um
satisfies (3.1) for m replaced by m− 1. From these facts and Frobenius theory again
it follows that
(3.3)
um+1 = −u0m + m cot θ um
and
(3.4)
[λ − (m − 1)(m + n − 2)] um−1 = u0m + (m + n − 2) cot θ um
if we agree to set c0 = 1 and define successive cm ’s via
λ − m(m + n − 1)
cm
(3.5)
for m = 0, 1, 2, . . . .
cm+1 =
2m + n
These constitute the raising and lowering relations for the functions um . Also,
elimination of u0m between (3.3) and (3.4) yields the pure recursion relation in m
(3.6)
um+1 − (2m + n − 2) cot θ um + [λ − (m − 1)(m + n − 2)] um−1 = 0.
Since (sin θ)m+n−2 um (θ) is 0 at θ = 0 one can integrate (3.4) and replace m by
m + 1 to obtain
Z θ
m+n−1
(3.7)
um+1 (θ) = [λ − m(m + n − 1)]
(sin t)m+n−1 um (t) dt.
(sin θ)
0
What we have developed so far could be considered the algebraic properties of the
functions um . One should think of the um ’s as the analogs of associated Legendre
functions in n dimensions. In particular, when n = 2 one finds that
um (θ; ν(ν + 1)) = (−1)m Pνm (cos θ)
following [1] (in general this should be correct up to a constant factor, typically a
factor of (−1)m , depending on the precise convention adopted; cf. [46], for example, where the convention differs by (−1)m ); here we have replaced the eigenvalue
parameter λ by ν(ν + 1), as is traditional in dealing with Legendre functions. In
this case (3.1) reduces to the associated Legendre equation and (3.3), (3.4), and
(3.6) all reduce to standard relations between the associated Legendre functions.
Almost certainly this generalization to n dimensions of associated Legendre functions and their basic relations has been developed before, though we do not know
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THE PPW CONJECTURE IN Sn
1065
of a reference where the details needed here are developed explicitly. (Cf. also [31]
which deals with the m = 0 case of (3.1) in n dimensions but in Schrödinger normal
form.) In any event, it is a relatively simple matter to pass to the n–dimensional
case once the situation in two dimensions is known. In fact, our functions um can be
expressed in terms of associated Legendre functions no matter what the dimension
but this connection is not particularly useful in this context so we do not elaborate
upon it here (but see the equations in Section 6 following equation (6.4), or Remark
1 below). We note that our convention on the constants cm allows them to vanish
from a certain value of m on for specific values of the parameter λ (cf. (3.5)). (Recall, for example, that when dealing with the full sphere S2 one needs P`m only for
` = 0, 1, . . . and m = 0, 1, . . . , `.) This is not a problem for our purposes here since
we will be most interested in passing from um to um+1 via either (3.3) or (3.7).
We come back now to the eigenvalue problem for (3.1). If one defines the lefthand side of equation (3.1) as the operator hm applied to y (with boundary conditions incorporated in the definition of hm ), then it is easily seen that hm0 > hm in
the sense of quadratic forms if m0 > m. Thus λ1 of −∆ for the polar cap is λ1 (h0 )
while λ2 of −∆ for the polar cap must be either λ1 (h1 ) or λ2 (h0 ). We now show
that the former is the case.
Lemma 3.1. The first eigenvalue of the Dirichlet Laplacian on a polar cap is the
first eigenvalue of (3.1) with m = 0 while the second eigenvalue of the Dirichlet
Laplacian is the first eigenvalue of (3.1) with m = 1. The second eigenvalue of
the cap occurs with multiplicity n. These results hold for all polar caps, i.e., for all
θ1 ∈ (0, π).
Proof. In the notation of the preceding paragraph we must show that λ2 (h0 ) >
λ1 (h1 ). The argument proceeds via a simple use of Rolle’s theorem as applied to
(3.3) and (3.4) rewritten in the forms
0
(3.8)
(sin θ)−m um = −(sin θ)−m um+1
and
(3.9)
(sin θ)m+n−2 um
0
= [λ − (m − 1)(m + n − 2)] (sin θ)m+n−2 um−1 .
In particular, with m = 0 in the first of these we have
(3.10)
u00 = −u1
and with m = 1 in the second we have
0
(3.11)
(sin θ)n−1 u1 = λ(sin θ)n−1 u0 .
By Rolle’s theorem, between any two zeros of u0 there is a zero of u00 and hence of
u1 , since (3.10) holds. Similarly, between two zeros of (sin θ)n−1 u1 there is a zero
of its derivative and hence, by (3.11), of u0 . Thus for fixed λ > 0 the zeros of u0
and (sin θ)n−1 u1 on [0, π) interlace.
Now consider u0 and u1 for λ = λ1 (h1 ). Since this makes θ1 the first positive zero
of u1 it is clear by what we have just proved that u0 has exactly one zero in (0, θ1 )
and that θ1 is not a zero of u0 . This then implies, by the fact that the positive
zeros of any um are decreasing functions of the parameter λ (see, for example, [20],
p. 315, or [28], p. 454), that λ2 (h0 ) > λ1 (h1 ).
That the multiplicity of λ1 (h1 ) as an eigenvalue of −∆ on the polar cap (=
geodesic ball) with Dirichlet boundary conditions is n follows from the details of
separation of variables. It can be shown that the corresponding eigenfunctions can
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1066
MARK S. ASHBAUGH AND RAFAEL D. BENGURIA
be taken as (xi / sin θ)y(θ) (restricted to Sn ) where i = 1, 2, . . . , n, xn+1 = cos θ,
and y(θ) is the eigenfunction of (3.1) for the eigenvalue λ1 (h1 ). These functions
form an orthogonal basis for the eigenspace of −∆ corresponding to the eigenvalue
λ1 (h1 ), showing that its multiplicity is n. This completes our proof.
Remarks. (1) The argument used in our proof above can be viewed as a way of
translating order properties of the zeros of the um ’s into order properties of the
corresponding eigenvalues λi (hm ). In fact, our approach extends easily to an interlacing result for the zeros (and hence for the associated eigenvalues) of um and um+1
for arbitrary m. Further ordering properties of the Dirichlet eigenvalues of spherical caps in Sn , at least for even n (and surely the case of odd n could be handled
similarly), may be inferred from the papers of Baginski [12], [13]. We note, though,
that all of Baginski’s results are presented in terms of the zeros of the associated
Legendre functions Pνm in the variable ν (for m an integer). To make the connection
to the present setting, one makes use of the formulas in Section 6 following equation
(6.4) (or equations (3.17) and (3.18) in [8], with the correction that the upper index
in both associated Legendre functions should be negated), which relate our functions um as defined above to associated Legendre functions (up to constant factors).
Note, in this connection, that y1 (θ) = u0 (θ; λ1 ) and y2 (θ) = u1 (θ; λ2 ) up to constant
factors (as proved in Lemma 3.1; throughout this section we take these factors to
−(n/2−1+m)
(cos θ),
be 1). In general, one has um (θ; λ) proportional to (sin θ)1−n/2 Pν
where λ and ν are related by λ = (ν − n/2 + 1)(ν + n/2). We thank the referee for
calling Baginski’s papers to our attention.
(2) Another proof of Lemma 3.1 follows by mimicking our proof of Lemma 3.1
of [9]. With τ = λ2 (h0 ) and v as the associated eigenfunction we can assume that
for some a ∈ (0, θ1 ) v > 0 on (0, a) and v < 0 on (a, θ1 ), and hence that v(a) = 0,
v 0 (a) < 0. Also we take λ = λ1 (h1 ) and set g = u1 (θ; λ) and h = u0 (θ; λ). Since
c1 = λ/n > 0 it is clear that g > 0 on (0, θ1 ) and that g(0) = 0 = g(θ1 ). It also
follows that g = −h0 and h satisfies
−h00 − (n − 1) cot θ h0 = λh
while v satisfies
−v 00 − (n − 1) cot θ v 0 = τ v.
From the last two equations we obtain
0
(sin θ)n−1 (vh0 − v 0 h) = −(λ − τ )(sin θ)n−1 vh
and integration from a to θ1 produces
(3.12) (sin θ1 )n−1 v 0 (θ1 )h(θ1 ) − (sin a)n−1 v 0 (a)h(a) = (λ − τ )
Z
θ1
vh sinn−1 θ dθ.
a
We now argue by contradiction, so assume τ ≤ λ. Since v(a) = 0, v = u0 (θ; τ ),
h = u0 (θ; λ) and τ ≤ λ, by the fact that the positive zeros of u0 (θ; λ) are decreasing
with increasing λ it follows that the first positive zero of h is less than or equal to
a and, since h0 = −g < 0 on (0, θ1 ), it therefore follows that h < 0 on (a, θ1 ]. But
now (3.12) gives a contradiction, since its right-hand side is greater than or equal
to 0 while its left-hand side is negative (note that v 0 (a) < 0 and v 0 (θ1 ) > 0 since a
and θ1 must be successive zeros of v).
(3) Yet another proof of Lemma 3.1 would be via the level–ordering results of
Baumgartner, Grosse, and Martin [18], [19]. Specifically, see our papers [3], [4]
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THE PPW CONJECTURE IN Sn
1067
where proofs for a ball in Rn occur and also the papers [16], [17] of Baumgartner,
which give extensions to cases arising from separation of variables in spherical
coordinates in spaces of constant curvature.
For future reference we note the following lemma.
Lemma 3.2. If 0 < θ1 < π, the first eigenfunction of (3.1) with m = 0, i.e.,
y1 (θ) ≡ u0 (θ; λ1 ), is strictly decreasing on [0, θ1 ] (y1 > 0 on [0, θ1 ) is our convention
for y1 here and throughout this paper; this follows from our choice c0 = 1).
Proof. u0 (θ; λ1 ) satisfies
−(sinn−1 θ u00 )0 = λ1 (θ1 ) sinn−1 θ u0 > 0
in [0, θ1 ) (since λ1 (Ω) > 0 follows from theR variational
R characterization of the
eigenvalues of −∆ via the Rayleigh quotient Ω |∇ϕ|2 / Ω ϕ2 ), which
implies that
sinn−1 θ u00 is decreasing in [0, θ1 ). Hence sinn−1 θ u00 < (sinn−1 θ u00 ) θ=0 = 0, which
proves the lemma.
Having identified λ1 and λ2 for −∆ on a spherical cap with Dirichlet boundary
conditions we are now in a position to investigate their behaviors and, in particular,
that of λ2 /λ1 . Since our concern will be with how these functions vary with θ1 ,
the geodesic radius of the spherical cap, we shall denote λ1 and λ2 by λ1 (θ1 ) and
λ2 (θ1 ) throughout the remainder of this section. Associated with equation (3.1) is
the one–dimensional Schrödinger operator
(3.13)
Hm (θ1 ) = −
d2
(2m + n − 1)(2m + n − 3) (n − 1)2
+
−
2
dθ
4
4 sin2 θ
acting on L2 ((0, θ1 ), dθ) with Dirichlet boundary conditions imposed at 0 and θ1 .
The operators Hm (θ1 ) form a family of self–adjoint operators. It is clear by Lemma
3.1 that λ1 (θ1 ) = λ1 (H0 (θ1 )) and λ2 (θ1 ) = λ1 (H1 (θ1 )).
We now analyze how λ1 and λ2 vary with θ1 by using perturbation theory [34],
[36], [45]. To be successful at this we need to work on a fixed interval (0, θ1 ) and
we do this by observing that the eigenvalue problem Hm (cθ1 )v = λv on (0, cθ1 ) can
be rescaled to
(3.14)
(2m + n − 1)(2m + n − 3) (n − 1)2
1 d2
−
v = λv
for t ∈ (0, θ1 )
− 2 2+
c dt
4
4 sin2 ct
which is equivalent to
(3.15)
(2m + n − 1)(2m + n − 3)c2
(n − 1)2 c2
d2
−
v = c2 λv
− 2+
dθ
4
4 sin2 cθ
for θ ∈ (0, θ1 ).
As was done above for Hm (θ1 ), we define an operator H̃m (c) on L2 (0, θ1 ) via the
differential expression appearing on the left-hand side of (3.15). It is clear from
(3.15) that λk (H̃m (c)) = c2 λk (Hm (cθ1 )) and thus, in particular, that
(3.16)
λ1 (cθ1 ) = c−2 λ1 (H̃0 (c))
and
(3.17)
λ2 (cθ1 ) = c−2 λ1 (H̃1 (c)).
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1068
MARK S. ASHBAUGH AND RAFAEL D. BENGURIA
What we intend to do now is to determine the derivatives λ01 (θ1 ) and λ02 (θ1 )
using perturbation theory and the fact that
1 dλj (cθ1 ) (3.18)
.
λ0j (θ1 ) =
θ1
dc
c=1
Since H̃m (c) is an analytic family in c for c near 1 we can apply regular Rayleigh–
Schrödinger perturbation theory [34], [36], [45]. In fact
c2
1
(2m + n − 1)(2m + n − 3)
−
H̃m (c) = Hm (θ1 ) +
4
sin2 cθ sin2 θ
(3.19)
(n − 1)2 (c2 − 1)
= Hm (θ1 ) + Vm (θ; c)
−
4
and since Vm (θ; c) is analytic in c for c near 1 we are assured that the operators
H̃m (c) form an analytic family of type (A) for c near 1 (see [45], p. 16 for the
definition of analytic family of type (A), or see [34], p. 154; also see Chapter 7
of [36] for the definitive account of analytic perturbation theory). This allows
us to compute the derivatives of the eigenvalues of H̃m (c) using the first-order
perturbation formula (cf. Kato [36], p. 391, eq. (3.18))
2
R θ1 ∂Vm
dλ1 (H̃m (c)) ∂c (θ; c) c=1 vm dθ
0
(3.20)
=
R θ1
2 dθ
dc
c=1
vm
0
where the functions vm (θ) denote first eigenfunctions of Hm (θ1 ) = H̃m (1).
By (3.18) we have
1 d λ2 (cθ1 )
d λ2 (θ1 )
=
dθ1 λ1 (θ1 )
θ1 dc λ1 (cθ1 )
c=1
d
d
1
1
λ2 (cθ1 ) λ1 (cθ1 ) =
.
λ1 (θ1 )
− λ2 (θ1 )
θ1
dc
dc
λ1 (θ1 )2
c=1
c=1
Thus, showing that λ2 (θ1 )/λ1 (θ1 ) increases with increasing θ1 comes down to showing that
d
d
1
1
λ2 (cθ1 ) λ1 (cθ1 ) (3.21)
−
,
0<
λ2 (θ1 ) dc
λ
(θ
)
dc
c=1
c=1
1 1
which, by equations (3.16) and (3.17), reduces to showing
d
d
1
1
λ1 (H̃1 (c)) λ1 (H̃0 (c)) (3.22)
−
.
0<
λ2 (θ1 ) dc
λ1 (θ1 ) dc
c=1
c=1
From (3.19) we obtain
1
1
∂Vm
(θ; c) = (2m + n − 1)(2m + n − 3) csc2 θ(1 − θ cot θ) − (n − 1)2
∂c
2
2
c=1
so that, by (3.20),
(3.23)
R θ1 (n + 1) csc2 θ(1 − θ cot θ) − (n − 1) v12 dθ
1
d
0
λ1 (H̃1 (c)) = (n − 1)
,
R θ1 2
dc
2
c=1
v dθ
0
1
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THE PPW CONJECTURE IN Sn
1069
and
(3.24)
R θ1 (n − 3) csc2 θ(1 − θ cot θ) − (n − 1) v02 dθ
1
d
0
.
λ1 (H̃0 (c)) = (n − 1)
R θ1 2
dc
2
c=1
v dθ
0
0
The functions v0 and v1 are related to u0 (θ; λ1 ) and u1 (θ; λ2 ) by v0 = u0 sin(n−1)/2 θ
and v1 = u1 sin(n−1)/2 θ respectively.
Introducing the functions `(θ) = cot θ − θ csc2 θ and m(θ) = −`0 (θ)/2 =
csc2 θ (1 − θ cot θ) we can write (3.23) as
R θ1
[(n + 1)m(θ) − (n − 1)] v12 dθ
1
d
(3.25)
,
λ1 (H̃1 (c)) = (n − 1) 0
R θ1 2
dc
2
c=1
v1 dθ
0
and (3.24) as
(3.26)
R θ1
[(n − 3)m(θ) − (n − 1)] v02 dθ
1
d
,
λ1 (H̃0 (c)) = (n − 1) 0
R θ1 2
dc
2
c=1
v dθ
0
0
0
respectively. Using the relations m(θ) = −` (θ)/2, `(θ) cot θ = m(θ) − 1, and
v02 = u20 sinn−1 θ, we can rewrite the numerator of the right-hand side of (3.26) as
Z θ1
[(n − 1)(m(θ) − 1) − 2 m(θ)] v02 dθ
0
Z
θ1
=
[(n − 1)`(θ) cot θ + `0 (θ)] u20 sinn−1 θ dθ
0
Z
(3.27)
θ1
=
[`(θ) sinn−1 θ]0 u20 dθ
0
Z
θ1
=
[−2`(θ) u0 (θ) u00 (θ)] sinn−1 θ dθ,
0
where the last equality follows by integrating by parts (both boundary terms vanish). Finally, from (3.26) and (3.27) we obtain
R θ1
[−`(θ) u0 (θ) u00 (θ)] sinn−1 θ dθ
d
λ1 (H̃0 (c)) = (n − 1) 0
(3.28)
.
R θ1 2
dc
c=1
0 v0 dθ
At this point we need the following properties of the functions `(θ) and m(θ).
Lemma 3.3. The function
`(θ) ≡ cot θ − θ csc2 θ
(3.29)
is negative, decreasing, and concave for 0 < θ < π. Moreover, the function
1
(3.30)
m(θ) ≡ − `0 (θ) = csc2 θ (1 − θ cot θ)
2
is positive, increasing, and convex for 0 < θ < π. Also, m(0) = 1/3, m(π/2) = 1,
and m(π − ) = ∞.
Q∞
Proof. Using the product representation of sin θ, i.e., sin θ = θ k=1 (1 − θ2 /(kπ)2 ),
one has
∞
∞
X
X
1
1
2
and
csc θ =
(3.31)
cot θ =
θ + kπ
(θ + kπ)2
k=−∞
k=−∞
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1070
MARK S. ASHBAUGH AND RAFAEL D. BENGURIA
(convergence of the series for cot θ here is understood in the sense of symmetric
partial sums). From (3.31) we obtain the following representation for `(θ):
∞ X
kπ
kπ
−
(3.32)
.
`(θ) = −
(kπ − θ)2
(kπ + θ)2
k=1
It follows from (3.32) that `(θ) < 0 for 0 < θ < π. Also from (3.32) we have
∞ kπ
kπ
`0 (θ) X
=
+
(3.33)
,
m(θ) = −
2
(kπ + θ)3
(kπ − θ)3
k=1
which is positive for 0 < θ < π. Thus, m(θ) is positive and `(θ) is decreasing for
0 < θ < π. Taking derivatives again, we find
∞ X
kπ
kπ
0
−
(3.34)
.
m (θ) = 3
(kπ − θ)4
(kπ + θ)4
k=1
The right-hand side of (3.34) is positive for θ ∈ (0, π). Hence, m(θ) is increasing
and `(θ) is concave in (0, π). It also follows from (3.34) that m0 (θ) is increasing,
and therefore m(θ) is convex in (0, θ1 ). Lastly, the values of m(θ) at θ = 0, π/2,
and π − are found by explicit evaluation.
Remark. In fact the function ` and all its derivatives are negative for 0 < θ < π.
With all these preliminary results in hand we are ready to prove Theorems 1.2
and 1.3.
Proof of Theorem 1.2. Since λ̃1 (c) ≡ λ1 (H̃0 (c)) = c2 λ1 (cθ1 ) (see (3.16) above), we
just need to prove that
1 d
d
λ̃1 (c) =
(3.35)
θ12 λ1 (θ1 ) < 0.
dc
θ1 dθ1
c=1
This inequality follows from (3.28) and Lemmas 3.2 and 3.3. Note that it holds for
all θ1 ∈ (0, π).
Proof of Theorem 1.3. Since λ2 (θ1 ) > λ1 (θ1 ) and dλ̃1 /dc c=1 < 0, to prove (3.22)
and therefore
λ2 (θ1 )
d
>0
dθ1 λ1 (θ1 )
reduces to showing (since λ̃1 λ̃02 − λ̃2 λ̃01 can be grouped as λ̃1 (λ̃02 − λ̃01 ) − (λ̃2 − λ̃1 )λ̃01 )
dλ̃1 dλ̃2 (3.36)
−
> 0,
dc c=1
dc c=1
where λ̃2 (c) ≡ λ1 (H̃1 (c)). From equations (3.25) and (3.26) this is equivalent to
proving
R θ1
R θ1
m(θ)v12 dθ
m(θ)v02 dθ
> (n − 3) 0 R θ1
,
(n + 1) 0 R θ1
2 dθ
2 dθ
v
v
1
0
0
0
which is obviously true for n ≤ 3 since m(θ) is positive. For n ≥ 4 it suffices to
show
R θ1
R θ1
2
m(θ)v02 dθ
0 m(θ)v1 dθ
> 0 R θ1
.
R θ1 2
v1 dθ
v02 dθ
0
0
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THE PPW CONJECTURE IN Sn
1071
The fact that g = v1 /v0 = y2 /y1 is an increasing function of θ on [0, θ1 ] for 0 <
Rθ
θ1 ≤ π/2 (see Section 4 below) implies that v̂1 ≡ v1 /( 0 1 v12 dθ)1/2 and v̂0 ≡
R θ1 2
v0 /( 0 v0 dθ)1/2 must have exactly one crossing in (0, θ1 ). Since m(θ) is positive
and increasing in [0, π), the desired inequality (3.36) follows by applying Lemma
2.7 of [14], p. 69; this inequality is also known as Bank’s inequality [15] (see also
[27] or [6], p. 607).
Remark. For n = 2 we can obtain a stronger version of Theorem 1.3 which holds
for all θ1 ∈ (0, π). Since m(θ) is increasing and m(0) = 1/3 we have (n + 1)m(θ) −
(n − 1) ≥ (n + 1)/3 − (n − 1) = 2(2 − n)/3 = 0 if n = 2. Thus, from (3.25) we have
dλ̃2 >0
dc c=1
if n = 2.
This inequality together with (3.35) implies (3.36) (or, even more directly, (3.22))
and therefore Theorem 1.3 for n = 2 and 0 < θ1 < π. In fact, this argument shows
that for all θ1 ∈ (0, π) θ12 λ1 (θ1 ) is decreasing and θ12 λ2 (θ1 ) is increasing. These
lead immediately to the fact that λ2 (θ1 )/λ1 (θ1 ) and θ12 [λ2 (θ1 ) − λ1 (θ1 )] are both
increasing for 0 < θ1 < π. For n = 3 a related argument allows us to show that
θ12 [λ2 (θ1 )−λ1 (θ1 )] is increasing for 0 < θ1 < π, and hence that Theorem 1.3 extends
to 0 < θ1 < π in that case as well. For n ≥ 4 we only have a proof of Theorem 1.3
for 0 < θ1 ≤ π/2.
Next we prove two inequalities between the first two Dirichlet eigenvalues of a
geodesic ball which are needed in Section 4.
Theorem 3.1. Let λ1 and λ2 be the first two eigenvalues of the Dirichlet Laplacian
on a geodesic ball contained in a hemisphere of Sn . Denote its (geodesic) radius by
θ1 . Then
(3.37)
n+2
λ2 − n
≥
λ1
n
for 0 < θ1 ≤ π/2
with equality if and only if θ1 = π/2 (i.e., for the hemisphere).
Remarks. (1) If we consider π/2 < θ1 < π, then inequality (3.37) is reversed (and is
a strict inequality, i.e., (λ2 − n)/λ1 < (n + 2)/n for π/2 < θ1 < π). This inequality
is actually quite interesting since it allows us to control the way λ2 goes to n (from
above) as θ1 → π − in terms of λ1 (since λ1 goes to 0 as θ1 → π − ). The proof
of this reversed inequality will not be given, since it follows by making suitable
modifications to the proof of Theorem 3.1, as given below.
(2) Theorem 3.1 is the analog for Sn of our Lemma 2.2 of [7] for the Euclidean
case. Indeed, the inequality of Lemma 2.2 follows (except that we do not get a
strict inequality) if we consider (3.37) in the limit θ1 → 0+ (the “Euclidean limit”).
Since λ1 ≈ α2 /θ12 and λ2 ≈ β 2 /θ12 for θ1 near 0 where α and β are the Bessel
function zeros jn/2−1,1 and jn/2,1 (see [1] for the notation here), the inequality
β 2 /α2 ≥ (n + 2)/n follows.
Proof. The proof is similar to that of Lemma 2.2 of [7]. We assume that θ1 < π/2
through most of the proof, returning to the case θ1 = π/2 (which can be treated
explicitly in terms of elementary functions) only at the end. We shall use a suitable
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1072
MARK S. ASHBAUGH AND RAFAEL D. BENGURIA
trial function (based on y2 ) in the Rayleigh quotient for λ1 = λ1 (θ1 ):
R θ1
R θ1 0 2 n−1
θ dθ
u(sinn−1 θ u0 )0 dθ
0 (u ) sin
(3.38)
= − 0R θ1
.
λ1 ≤ R θ1
u2 sinn−1 θ dθ
u2 sinn−1 θ dθ
0
0
It suffices that the trial function u be real and continuous on [0, θ1 ], have u(0) finite
and u(θ1 ) = 0, and be such that all the integrals occurring above exist as finite real
numbers. This includes, in particular, the case u = y2 / sin θ, which we now adopt.
With this choice we find
y 0 0
d
2
= −
sinn−1 θ
− sinn−1 θ u0
dθ
sin θ
(3.39)
= − sinn−2 θ y200 − (n − 3) sinn−2 θ cot θ y20
+ (n − 3) sinn−4 θ y2 − (n − 2) sinn−2 θ y2 ,
and, upon using the differential equation satisfied by y2 (i.e., (3.1) with m = 1), we
have
(3.40)
0
− sinn−1 θ u0 = 2 sinn−2 θ cot θ y20 + (λ2 − n + 2) sinn−2 θ y2 − 2 sinn−4 θ y2 .
That the expression on the right here has a finite limit as θ → 0+ follows from the
fact that y2 (θ) can be taken as u1 (θ) and u1 (θ) = c1 θ + O(θ3 ) as θ → 0+ (see (3.1)
and (3.2)). It now follows from (3.38) and (3.40) (since for θ1 < π/2, u = y2 / sin θ
does not satisfy the equation that y1 does, we can write a strict inequality here)
that
Z θ1
uy2 sinn−2 θ dθ
λ1
0
(3.41)
Z θ1
u 2 cot θ y20 + (λ2 − n + 2)y2 − 2 csc2 θ y2 sinn−2 θ dθ.
<
0
Since we would like to show that λ1 < n(λ2 − n)/(n + 2), it will be enough to show
that the right-hand side of (3.41) is less than or equal to
Z θ1
uy2 sinn−2 θ dθ,
[n(λ2 − n)/(n + 2)]
0
or, equivalently,
Z θ1
u 2 cot θ y20 + (λ2 − n + 2)y2 − 2 csc2 θ y2 sinn−2 θdθ
(n + 2)
0
Z
θ1
≤ n(λ2 − n)
uy2 sinn−2 θdθ
0
or
(3.42)
Z
θ1
2
u (n + 2) cot θ y20 + (λ2 + 2)y2 − (n + 2) csc2 θ y2 sinn−2 θdθ ≤ 0.
0
We now rewrite this integral using y2 = u1 (θ; λ2 ) (where λ2 = λ2 (θ1 )) and proceed
to simplify the integrand using the relations (3.3), (3.6) developed for the um ’s.
First, using (3.3) with m = 1 the expression in square brackets becomes
(n + 2) cot θ[cot θ u1 − u2 ] + (λ2 + 2)u1 − (n + 2) csc2 θ u1
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THE PPW CONJECTURE IN Sn
1073
or, since cot2 θ − csc2 θ = −1,
−(n + 2) cot θ u2 + (λ2 − n)u1 .
Finally, we use the recursion relation (3.6) with m = 2 to see that
u3 = (n + 2) cot θ u2 − (λ − n)u1
and hence, since we are taking λ = λ2 , inequality (3.42) may be rewritten simply
as
Z θ1
uu3 sinn−2 θ dθ ≤ 0,
−2
0
which clearly holds if u3 > 0 for 0 < θ < θ1 since u = y2 / sin θ and y2 > 0 on
(0, θ1 ). To see that u3 > 0 for 0 < θ < θ1 we use the relation (3.7), first for m = 1
and then for m = 2. With m = 1 we have
Z θ
(sin t)n u1 (t) dt,
sinn θ u2 (θ) = (λ2 − n)
0
showing that u2 (θ; λ2 ) > 0 for 0 < θ ≤ θ1 since u1 (θ; λ2 ) = y2 (θ) > 0 on (0, θ1 ) and
λ2 = λ2 (θ1 ) > n for 0 < θ1 ≤ π/2 by domain monotonicity of Dirichlet eigenvalues
(and the fact that λ2 (π/2) = 2(n+1) > n). This in turn yields u3 > 0 for 0 < θ ≤ θ1
if 0 < θ1 < π/2 since by (3.7) with m = 2 we have
Z θ
n+1
θ u3 (θ) = (λ2 − 2(n + 1))
(sin t)n+1 u2 (t) dt,
sin
0
and we know by domain monotonicity that λ2 (θ1 ) > λ2 (π/2) = 2(n + 1) for 0 <
θ1 < π/2.
Thus we have proved inequality (3.37) of our theorem for 0 < θ1 < π/2 (with
strict inequality in (3.37)). It only remains to show that equality holds when
θ1 = π/2 and this is elementary since in this case we can solve explicitly for all
eigenvalues and eigenfunctions without even the need of special functions. One
finds that λ1 = n with eigenfunction xn+1 = cos θ and that λ2 = 2(n + 1) with
exactly n linearly independent eigenfunctions xi xn+1 for i = 1, 2, . . . , n. Clearly
(λ2 − n)/λ1 = (n + 2)/n and our proof is complete.
Remark. To fill in the picture for the hemisphere in Sn , we note that y1 = cos θ
and y2 = sin θ cos θ, and thus u1 = sin θ cos θ, u2 = sin2 θ, and u3 ≡ 0. In this case
u = y2 / sin θ ≡ y1 and (3.38) becomes an equality.
Lemma 3.4. With notation as above,
(3.43)
λ2 − λ1 >
n−1
sin2 θ1
for 0 < θ1 ≤ π/2.
Proof. The proof is similar to that of Lemma 2.1 of [7]. We use u = y2 in the
Rayleigh quotient for λ1 (i.e., in (3.38) above). Since
n−1
n−1
0 0
n−1
θ y2 = sin
θ λ2 −
y2 ,
− sin
sin2 θ
we get from (3.38)
R θ1 2
u ((n − 1)/ sin2 θ) sinn−1 θ dθ
(3.44)
λ1 < λ2 − 0
R θ1
u2 sinn−1 θ dθ
0
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1074
MARK S. ASHBAUGH AND RAFAEL D. BENGURIA
(again, y2 does not satisfy the equation that y1 does, so (3.44) is strict). Now the
lemma follows by the monotonicity of sin θ in (0, θ1 ) for 0 < θ1 ≤ π/2.
To conclude this section we present the following results which are needed in
Section 4.
Theorem 3.2. Let p(θ) = −u00 (θ, λ1 )/u0 (θ, λ1 ) = u1 (θ, λ1 )/u0 (θ, λ1 ). Then, p(θ)
is positive, strictly increasing, and strictly convex on (0, θ1 ) for 0 < θ1 ≤ π/2.
Moreover, p(0) = 0 and p(θ) → ∞ as θ → θ1− .
Proof. The analog of this result in the Euclidean case was proved in Lemma 2.3 of
[7]. That p(0) = 0 and p(θ) → ∞ as θ → θ1− follow from the boundary behavior of
u0 . Using the raising and lowering identities (3.3) and (3.4) with m = 0 and m = 1
respectively and with λ fixed at λ1 (θ1 ), i.e.,
u1 = −u00 ,
(3.45)
λ1 u0 = u01 + (n − 1) cot θ u1 ,
one obtains
(3.46)
u20 p0 = λ1 u20 + u21 − (n − 1) cot θ u0 u1 ≡ σ(θ),
and in similar fashion
(3.47)
0
sinn−1 θ σ(θ) = (n − 1) sinn−3 θ u0 u1 .
The function σ(θ) is finite at θ = 0, and thus σ(θ) sinn−1 θ = 0 at θ = 0. Since
u0 is positive and decreasing in (0, θ1 ), u0 u1 = −u0 u00 > 0 there. Hence, σ(θ) > 0
in (0, θ1 ] and therefore by (3.46) p is increasing there. Clearly p(θ) > 0 in (0, θ1 ),
since p(0) = 0 and p is increasing there. Moreover, from (3.46) we find
u20 p
− (n − 1) cot θ σ + 2pσ ≡ s(θ).
sin2 θ
From (3.45), (3.46), (3.47), and (3.48) we obtain
0
σ
(3.49)
sin
θ
+
(n
−
1)
sin
θ
+
2p
cos
θ
>0
sinn+1 θ s(θ) = 2 sinn θσ(θ)
u20
(3.48)
u20 p00 = (n − 1)
in (0, θ1 ) for 0 < θ1 ≤ π/2. Since sinn+1 θ s(θ) = 0 at θ = 0, this implies that s(θ) >
0 in (0, θ1 ) and therefore, by (3.48), p00 > 0 there and the proof is complete.
Remark. The function p satisfies the Riccati equation
(3.50)
p0 = λ1 + p2 − (n − 1) cot θ p.
An alternative proof of Theorem 3.2 can be given directly from (3.50). In particular,
the fact that p is increasing follows from (3.50) and the convexity of cot θ in (0, π/2)
using arguments similar to the ones used to prove Theorem 4.1 below.
Lemma 3.5. The function p(θ) cot θ is strictly increasing on (0, θ1 ) for 0 < θ1 <
π/2. If θ1 = π/2, p(θ) = tan θ, so p(θ) cot θ ≡ 1 in that case.
Proof. Consider the function r(θ) ≡ p(θ) cot θ − λ1 /n. Using (3.47), (3.48), and
(3.50) one can show that r(θ) satisfies the equation
0
(3.51)
sinn−1 θ u20 r0 = 2n sinn−3 θ u20 r.
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THE PPW CONJECTURE IN Sn
1075
Since the function p(θ) is odd and analytic in θ for θ near 0, we can expand it in
odd powers of θ. Inserting a series expansion in the Riccati equation (3.50) we find
p(θ) =
λ1
λ1
θ+ 2
(3λ1 + n(n − 1)) θ3 + O(θ5 )
n
3n (n + 2)
for θ near 0. Substituting this into the definition of r(θ) we find
r(θ) ≈
λ1
(λ1 − n)θ2
n2 (n + 2)
as θ → 0+ . Since λ1 (θ1 ) > n for θ1 < π/2 (which follows from the fact that
λ1 (π/2) = n and λ1 (θ1 ) is decreasing in θ1 ), r(θ) is positive in a neighborhood of
0. Now, r(θ) is a continuously differentiable function in [0, θ1 ), it is positive in a
neighborhood of 0, and it goes to infinity as θ → θ1− . This implies that either r(θ) is
an increasing function in [0, θ1 ) or that it has a positive local maximum. However,
this latter possibility is ruled out by (3.51) (any possible positive critical point of
r(θ) must be a local minimum). Thus, r(θ) is increasing on (0, θ1 ) if 0 < θ1 < π/2.
If θ1 = π/2, u0 = cos θ, hence u00 = − sin θ, p(θ) = tan θ, and finally r(θ) ≡ 0 since
λ1 = n.
Remark. In the Euclidean case, the analog of Lemma 3.5 (i.e., Lemma 2.4 of [7])
was an immediate consequence of the convexity of the function analogous to the
function p(θ) used here. For Sn , as we have seen, the proof is somewhat more
involved.
4. Monotonicity properties of g and B
In this section we prove the key properties of the functions that occur in our
rearrangement procedure in Section 6. These functions are
(4.1)
q(θ) = sin θ
and
(4.2)
B(θ) = g 0 (θ)2 +
g 0 (θ)
,
g(θ)
g 2
n−1
2
2
=
[q
+
(n
−
1)]
,
g(θ)
sin θ
sin2 θ
where
(4.3)
g(θ) =
u1 (θ; λ2 (θ1 ))
y2 (θ)
=
.
y1 (θ)
u0 (θ; λ1 (θ1 ))
Our objectives in this section are to prove that g(θ) is increasing and B(θ) is
decreasing in [0, θ1 ] for 0 < θ1 ≤ π/2. For the hemisphere (i.e., for θ1 = π/2) we
can explicitly compute g(θ) = sin θ and B(θ) = (n − 1) + cos2 θ (see the remark
following the proof of Theorem 3.1). It is obvious that g is increasing and B
is decreasing in this case. Thus, we can assume in the rest of this section that
θ1 < π/2.
Since g and sin θ are positive, that g is increasing will be a simple consequence
of showing q ≥ 0. On the other hand, from (4.1) and (4.2) it follows that
2
g(θ)
1
0
0
2
(4.4)
.
B (θ) = 2 qq − (cos θ − q)(q + n − 1)
sin θ
sin θ
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1076
MARK S. ASHBAUGH AND RAFAEL D. BENGURIA
Hence, that B is decreasing will be a consequence of showing that q 0 ≤ 0 and
0 ≤ q ≤ cos θ. Thus, in order to prove the desired properties of g and B we only
need to show that 0 ≤ q(θ) ≤ cos θ and q 0 (θ) ≤ 0 for 0 ≤ θ ≤ θ1 .
The strategy we use to prove these results for q follows the same general method
used in [7] and [9]. Since we shall need the boundary behavior of q at the two
endpoints θ = 0 and θ = θ1 (this is necessitated by the fact that the coefficients in
the right-hand side of the differential equation for q, equation (4.7) below, become
singular at the two endpoints), we give these now. By Taylor–Frobenius expansion,
we find
λ1
λ2
2−n
−
−
q 00 (0) = 2
(4.5)
,
q(0) = 1,
q 0 (0) = 0,
n
n + 2 2(n + 2)
(4.6)
q(θ1 ) = 0,
q 0 (θ1 ) = −
1
3
n−1
λ2 − λ1 −
sin θ1 .
sin2 θ1
Throughout this section we will use λ1 and λ2 to denote λ1 (θ1 ) and λ2 (θ1 ), respectively. From (4.5) and Theorem 3.1 we have that q 00 (0) < −1 for θ1 < π/2. From
(4.6) and Lemma 3.4 we have that q 0 (θ1 ) < 0. Therefore, q < cos θ on an interval
just to the right of 0 and q > 0 on an interval just to the left of θ1 .
In order to prove that q ≥ 0, q 0 ≤ 0, and q ≤ cos θ for 0 ≤ θ ≤ θ1 we analyze
the ordinary differential equation satisfied by q. To obtain the differential equation
for q first we differentiate (4.1) with respect to θ, and use our choice of g = y2 /y1 .
Then we use the equation (3.1), with m = 0 and λ = λ1 (θ1 ), satisfied by y1 , and
the same equation, but this time with m = 1 and λ = λ2 (θ1 ), satisfied by y2 . Thus,
we obtain
(4.7)
q 0 = 2pq − (n − 2)q cot θ −
q2 + 1 − n
− (λ2 − λ1 ) sin θ.
sin θ
Here, the function p ≡ −y10 /y1 obeys the Riccati equation
(4.8)
p0 − p2 + (n − 1) cot θ p − λ1 = 0
associated to equation (3.1) with m = 0 and λ = λ1 (θ1 ) (note that (4.7) is also
a Riccati equation). We proved in Section 3 that p is positive, strictly increasing,
and strictly convex in (0, θ1 ), for 0 < θ1 ≤ π/2, p(0) = 0, and p → ∞ as θ → θ1−
(see Theorem 3.2).
Having derived the equation for q, we are ready to prove the necessary facts
about q. We start by showing q ≥ 0 in [0, θ1 ], which we prove by contradiction.
Assume q is negative somewhere in [0, θ1 ]. Since q(0) = 1 and q is positive to the
left of θ1 (and q is continuous) this implies that there are two points α, β, say, with
0 < α < β < θ1 such that q(α) = q(β) = 0 and q 0 (α) ≤ 0, q 0 (β) ≥ 0. At points in
(0, θ1 ) where q = 0 it follows from (4.7) that
(4.9)
q0 =
n−1
− (λ2 − λ1 ) sin θ.
sin θ
Since λ2 > λ1 and sin θ is increasing in θ for 0 < θ < π/2, the right-hand side of
(4.9) is strictly decreasing in θ, and hence it is not possible to have α < β with
q(α) = q(β) = 0 and q 0 (α) ≤ 0, q 0 (β) ≥ 0. Therefore q ≥ 0 in [0, θ1 ].
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THE PPW CONJECTURE IN Sn
1077
The proof that q ≤ cos θ follows the same ideas. Define the function ψ = cos θ−q.
From (4.7) we get
ψ
(4.10)
(ψ − cos θ) − (n − 1)ψ cot θ.
ψ 0 = (λ2 − λ1 − n) sin θ + 2 p +
sin θ
We have already shown that ψ(θ) = cos θ − q(θ) is positive in a neighborhood of
θ = 0. Also, if θ1 < π/2, ψ(θ1 ) = cos θ1 > 0. Now assume that ψ is negative
somewhere in [0, θ1 ]. This implies that there are two points r, s, say, with 0 < r <
s < θ1 such that ψ(r) = ψ(s) = 0 and ψ 0 (r) ≤ 0, ψ 0 (s) ≥ 0. At points in (0, θ1 )
where ψ = 0 we have from (4.10) that
1
ψ 0 = (λ2 − λ1 − n) − 2 p cot θ.
sin θ
From Lemma 3.5 it follows that the right-hand side of (4.11) is strictly decreasing
in θ, hence it is not possible to have r < s with ψ(r) = ψ(s) = 0 and ψ 0 (r) ≤ 0,
ψ 0 (s) ≥ 0. Thus ψ ≥ 0, and hence q ≤ cos θ in [0, θ1 ].
Finally, we show that q(θ) is decreasing in [0, θ1 ]. It is convenient now to write
(4.7) in an alternative form and use convexity arguments. Specifically we consider
(4.11)
q 0 = 2 p(θ) q + (n − 2)(1 − q) cot θ
(4.12)
1 − q2
+ (n − 2) tan(θ/2) − (λ2 − λ1 ) sin θ
sin θ
and, since λ2 − λ1 > 0, q ≤ 1, and n ≥ 2, the right-hand side of (4.12), F (θ, q),
say, is convex in θ for fixed q because each of the functions cot θ, csc θ, tan(θ/2),
and − sin θ is individually convex on the interval [0, θ1 ] for 0 < θ1 ≤ π/2. Also the
function p(θ) is convex on the same interval (see Theorem 3.2 above). To see how
these facts imply that q 0 ≤ 0 observe that if not we could find three points α1 , α2 ,
α3 in (0, θ1 ), where q(α1 ) = q(α2 ) = q(α3 ), q 0 (α1 ) < 0, q 0 (α2 ) > 0, and q 0 (α3 ) < 0.
But then we would have (we use q for the common value of q(αi ) here)
+
(4.13)
0 < q 0 (α2 ) = F (α2 , q) = F (µ α1 + (1 − µ)α3 , q)
< µF (α1 , q) + (1 − µ)F (α3 , q) = µq 0 (α1 ) + (1 − µ)q 0 (α3 ) < 0,
a contradiction. We have used the strict convexity of F (θ, q) in θ here. The parameter µ is strictly between 0 and 1 and determines α2 as a convex combination
of α1 and α3 , that is, α2 = µ α1 + (1 − µ)α3 . To summarize this section we state
the results above as a theorem.
Theorem 4.1. With q(θ), B(θ), and g(θ) as defined in (4.1), (4.2), and (4.3),
respectively, the inequalities 0 ≤ q ≤ cos θ and q 0 ≤ 0 hold on [0, θ1 ] for 0 < θ1 ≤
π/2. It follows that g(θ) is increasing and B(θ) is decreasing there as well.
5. Chiti’s comparison argument in Sn
Here we need an extension of Chiti’s comparison result [24], [25], [26], [27] (see
also Appendix A of [6]), given originally for domains in Rn , to the case of domains
in Sn .
We let Sn (ρ) denote the n–dimensional sphere of radius ρ (hence of constant
sectional curvature κ = 1/ρ2 ). Sn will always denote Sn (1). Define the function
Sρ (r) = ρ sin(r/ρ).
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1078
MARK S. ASHBAUGH AND RAFAEL D. BENGURIA
It is well known that, in geodesic polar coordinates, the metric on Sn (ρ) is
2
ds2 = dr2 + Sρ (r)2 |dω| ,
2
where r represents geodesic distance from a point and |dω| is the canonical metric
for Sn−1 . One should think of r as ρ times the angle θ from the north pole. Thus
r runs from 0 to ρπ.
We will always assume that r = 0 (the north pole) is the center for our spherical
rearrangements (this is not a restriction since the metric is identical in geodesic
polar coordinates about any point). Then for a bounded domain Ω ⊂ Sn (ρ) Ω? ,
the spherical rearrangement of Ω, will denote the geodesic ball about r = 0 having
the same volume as Ω, i.e., |Ω? | = |Ω|.
For a nonnegative function f defined on Ω we define two rearranged functions f #
and f ? . The decreasing rearrangement f # of f is a function from [0, |Ω|] to R which
is equimeasurable with f and nonincreasing. We will use s as the argument of f # in
most instances. The symmetric decreasing (or spherical decreasing) rearrangement
f ? of f is a function defined on Ω? which is invariant under rotations about r = 0,
equimeasurable with f , and nonincreasing with respect to r. f ? is a function of
x ∈ Ω? but because of its symmetry we will abuse notation and write f ? (r) where
r is the geodesic distance from the center of Ω? . With this understanding we have
f ? (r) = f # (A(r))
where
(5.1)
Z
r
A(r) ≡ s = nCn
Sρ (τ )n−1 dτ
0
is the n-volume of the geodesic ball of radius r in Sn (ρ). Here Cn = π n/2 /Γ( n2 + 1)
denotes the volume of the unit ball in Rn (and nCn is its “surface area”, i.e.,
|Sn−1 |). We shall also have occasion to use increasing rearrangements. These
will be denoted f# (the increasing rearrangement of f ) and f? (the symmetric or
spherical increasing rearrangement of f ) and their definitions are analogous to those
of f # and f ? , respectively. For further information on rearrangements the reader
is referred to the book of Hardy, Littlewood, and Pólya [32] and the many other
references given in [6], [51].
Finally, we also need the classical isoperimetric inequality extended to Sn (ρ) in
its sharp form. We let L(t) be the function giving the (n − 1)–dimensional volume
of the geodesic ball of radius r, i.e.,
(5.2)
L(r) = nCn Sρ (r)n−1 = A0 (r).
Then, for example, for any domain in S2 one has
2
A
2
(5.3)
= 4πA − κA2
L ≥ 4πA −
ρ
(see, e.g., [41]). Here A = |Ω| and L is the length of the boundary of Ω. Equality
occurs in (5.3) if and only if Ω is a geodesic ball. In Sn , n > 2, the sharp classical
isoperimetric inequality cannot be given in as explicit a form as (5.3). For a bounded
domain Ω we define Hn−1 (∂Ω) to be the (n − 1)–dimensional volume of ∂Ω. The
isoperimetric inequality on Sn (ρ) then reads
(5.4)
Hn−1 (∂Ω) ≥ Hn−1 (∂Ω? )
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THE PPW CONJECTURE IN Sn
1079
with equality if and only if Ω is a geodesic ball (see Burago and Zalgaller [21], p. 86,
Theorem 10.2.1). In terms of the function L(r) defined above (5.4) may be written
(5.5)
Hn−1 (∂Ω) ≥ L(θ(|Ω|)) = nCn Sρ (θ(|Ω|))n−1 ,
where θ(s) is the inverse function to the function A defined in (5.1).
With all these ingredients we state our extension of Chiti’s comparison result
(throughout the rest of this section we set ρ = 1, the extension to arbitrary ρ being
straightforward).
Theorem 5.1. Let Ω be a bounded domain in Sn and let λ1 and u1 denote the
first Dirichlet eigenvalue and eigenfunction of the Laplacian on Ω. Let Bλ1 be the
geodesic ball of such a radius that λ1 is also the first Dirichlet eigenvalue of the
Laplacian on Bλ1 . LetRv1 > 0 be the
R first Dirichlet eigenfunction on Bλ1 and fix its
normalization so that Ω u21 dσ = Bλ v12 dσ.
1
Then there is a value r1 ∈ (0, θ(|Bλ1 |)) such that
(5.6)
v1 (r) ≥ u?1 (r)
for r ∈ [0, r1 ],
and
(5.7)
v1 (r) ≤ u?1 (r)
for r ∈ [r1 , θ(|Bλ1 |)],
where θ(s) is the function defined following (5.5) above.
Proof. Let u1 , respectively λ1 , be the lowest eigenfunction, respectively eigenvalue,
of the Dirichlet problem on Ω ⊂ Sn , i.e.,
in Ω,
u1 = 0
on ∂Ω.
−∆u1 = λ1 u1
Define Ωt = {x u1 (x) > t} and ∂Ωt = {x u1 (x) = t}. Let µ1 (t) = |Ωt |, and
|∂Ωt | ≡ Hn−1 (∂Ωt ), where Hn−1 (dσ) denotes (n − 1)–dimensional measure on Sn .
Then we have (see, e.g., Talenti [51], p. 709, eq. (32))
Z
1
0
Hn−1 (dσ),
(5.9)
−µ1 (t) =
|∇u
1|
Ωt
(5.8)
for almost every t > 0. Applying Gauss’s theorem to (5.8), we have
Z
Z
(5.10)
u1 dσ =
|∇u1 |Hn−1 (dσ),
λ1
Ωt
∂Ωt
since the outward normal to Ωt is given by −∇u1 /|∇u1 |. Using the Cauchy–Schwarz
inequality and equations (5.9) and (5.10) we obtain
Z
2
Z
2
0
(5.11)
Hn−1 (dσ) ≤ (−µ1 (t))λ1
u1 dσ.
|∂Ωt | =
∂Ωt
Ωt
As discussed above, if Ω is a domain in S , the classical isoperimetric inequality is
given by
n
(5.12)
Hn−1 (∂Ω) ≥ Hn−1 (∂Ω? )
where Ω? is a geodesic ball having the same volume as Ω. The (n − 1)–dimensional
measure of ∂Ω? , Hn−1 (∂Ω? ), is given in terms of θ1 (Ω? ), the geodesic radius of Ω? ,
by
(5.13)
Hn−1 (∂Ω? ) = nCn (sin θ1 (Ω? ))
n−1
,
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1080
MARK S. ASHBAUGH AND RAFAEL D. BENGURIA
where Cn = π n/2 /Γ( n2 + 1) is the volume of the unit ball in Rn (and nCn is the
volume of Sn−1 ). Therefore, from (5.12) and (5.13) it follows that
(5.14)
|∂Ωt | = Hn−1 (∂Ωt ) ≥ nCn (sin θ1 (Ω?t ))
n−1
.
Hence, from (5.11) we have
Z
1
2n−2
u1 dσ ≥ n2 Cn2 [sin θ1 (Ω?t )]
(5.15)
.
− 0
λ1
µ1 (t)
Ωt
Finally one uses the fact that
Z
Z
(5.16)
u1 dσ =
Ωt
µ1 (t)
u#
1 (s) ds,
0
which follows directly from the definition of u#
1 , the decreasing rearrangement of
u1 on the interval [0, |Ω|] (here s is a variable denoting volume and is related to the
Rθ
geodesic radial variable θ via s = nCn 0 (sin r)n−1 dr, i.e., ds/dθ = nCn (sin θ)n−1 ).
Since u#
1 (s) is the inverse function to µ1 (t), we have
−
1
du#
1
=− 0 ,
ds
µ1 (t)
which, combined with (5.15) and (5.16), yields
(5.17)
du#
− 1 ≤ λ1 n−2 Cn−2 (sin θ(s))2−2n
ds
Z
s
0
0
u#
1 (s ) ds .
0
Now, if we view v1 as a function of the volume s (here we will abuse notation and continue to call it v1 ) rather than as a function of θ (or r), with s =
Rθ
nCn 0 (sin r)n−1 dr, it satisfies (5.17) with equality, i.e.,
Z s
dv1
= λ1 n−2 Cn−2 (sin θ(s))2−2n
(5.18)
v1 (s0 ) ds0 .
−
ds
0
In fact, (5.18) is an integrated version of equation (3.1) with m = 0 and λ = λ1
in the variable s. Having obtained the relations (5.17) and (5.18) for u#
1 (s) and
v1 (s) respectively, we will prove that under the normalization condition imposed on
them, they are either identical or they cross only once in the interval (0, |Bλ1 |) (in
the sense specified by Theorem 5.1). All the arguments we give below depend on
the continuity of u#
1 and v1 . The function v1 is in fact real analytic in [0, |Bλ1 |] and,
furthermore, it is decreasing there as can be seen from Lemma 3.2 above (or from
(5.18) and the fact that v1 > 0). The absolute continuity of u#
1 on [0, |Ω|] follows
#
from arguments in [51]. Since u1 and v1 are normalized to have the same L2 –
norm, they either are identical or they cross. If they coincide, then Bλ1 = Ω? , and
Theorem 5.1 is proved since any r ∈ (0, θ(|Bλ1 |)) will serve as r1 . Next, following
#
Chiti [26] we conclude that u#
1 (0) cannot exceed v1 (0). In fact, if u1 (0) ≥ v1 (0)
it follows by mimicking the proof of the main theorem in [26] that v1 (s) ≤ u#
1 (s)
#
for all s ∈ [0, |Bλ1 |) which, in turn, implies v1 (s) ≡ u1 (s) and we are back in the
#
previous case. Thus, if v1 (s) 6≡ u#
1 (s), v1 (s) > u1 (s) in a neighborhood of 0, and
both functions being of the same norm they must cross at least once. Let s1 be
0
0
0
the largest s ∈ (0, |Bλ1 |) such that u#
1 (s ) ≤ v1 (s ) for all s ≤ s. By the definition
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THE PPW CONJECTURE IN Sn
1081
of s1 , there is an interval immediately to the right of s1 on which u#
1 (s) > v1 (s).
Indeed, by continuity and the definition of s1
Z s
0
0
0
(5.19)
[u#
for 0 < s ≤ s1 + 1 (s ) − v1 (s )] ds < 0
0
for some > 0. It now follows that u#
1 (s) > v1 (s) at least on the interval from s1
to s1 + since by the absolute continuity of u#
1
Z s
d
(v1 − u#
v1 (s) − u#
1 (s) =
1 ) ds
s1 ds
Z s
Z s0
(5.20)
−2 −2
0 2−2n
00
00
00
0
(sin θ(s ))
[u#
≤ λ1 n Cn
1 (s ) − v1 (s )] ds ds
0
s1
<0
for s ∈ (s1 , s1 + ]
by virtue of (5.19).
We will now show that u#
1 (s) > v1 (s) for all s ∈ (s1 , |Bλ1 |], which will prove the
theorem. If not, then the point s2 defined to be the largest s ∈ (s1 , |Bλ1 |] for which
0
0
0
u#
1 (s ) > v1 (s ) for all s1 < s < s, would be less than |Bλ1 |, and we would have
u#
1 (s) > v1 (s)
(5.21)
for s ∈ (s1 , s2 )
#
with u#
1 (s1 ) = v1 (s1 ) and u1 (s2 ) = v1 (s2 ). In this case, we can define the function
(
v1 (s) for s ∈ [0, s1 ] ∪ [s2 , |Bλ1 |],
w(s) =
u#
1 (s) for s ∈ (s1 , s2 ).
It follows from (5.17) and (5.18) that w satisfies
dw
− (s) ≤ λ1 n−2 Cn−2 (sin θ(s))2−2n
ds
(5.22)
Z
s
w(s0 ) ds0 ,
0
with strict inequality for all s > s1 . From w(s) define the function
g(x) = w(s(θ))
for x ∈ Bλ1 where θ is the polar angle (angle from the north pole) corresponding to
x. Thus g is a radial function on Bλ1 (assumed centered at the norh pole). Because
of (5.22) (or (5.20) or (5.21)), g cannot be the groundstate of the Laplacian with
Dirichlet boundary conditions on Bλ1 (but it is certainly an admissible trial function
for λ1 ). Therefore,
R
2
Bλ1 |∇g| dσ
R
(5.23)
.
λ1 <
g 2 dσ
Bλ
1
By a standard change of variables,
Z
Z
2
(5.24)
g dσ =
Bλ1
and
|∇g|2 dσ = n2 Cn2
Bλ1
w2 (s) ds
0
Z
Z
(5.25)
|Bλ1 |
|Bλ1 |
(sin θ(s))2n−2 w0 (s)2 ds.
0
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1082
MARK S. ASHBAUGH AND RAFAEL D. BENGURIA
Using (5.22) (substitute for one of the w0 (s)’s in (5.25), using the fact that −w0 (s) >
0) and integration by parts we get
Z |Bλ1 |
Z
(5.26)
|∇g|2 dσ ≤ λ1
w(s)2 ds.
Bλ1
0
From (5.23), (5.24), and (5.26) we get a contradiction, and the theorem follows.
6. The main result
After all the preliminaries developed in Sections 2 through 5 we are ready to
prove our main result, i.e., Theorem 1.1 from which the PPW result for domains
contained in a hemisphere of Sn follows as indicated in the introduction. As in our
proof of the PPW conjecture for domains in Rn , the starting point here is the use
of the gap inequality, which is a variational estimate for the difference between the
first two eigenvalues of the Laplacian. The gap inequality states that
R
2
|∇P | u21 dσ
(6.1)
λ2 (Ω) − λ1 (Ω) ≤ ΩR
2 2
Ω P u1 dσ
R
provided Ω P u21 dσ = 0 and P 6≡ 0. Here Ω is a domain in Sn and dσ is the
standard volume element in Sn . The gap inequality follows from the Rayleigh–Ritz
inequality for λ2 using P u1 as the trial function (hence the side condition P u1 ⊥ u1 )
after a suitable integration by parts. To get the desired isoperimetric result out of
this, one must make very special choices of the function P ; in particular, choices
such that (6.1) is an equality if Ω is the appropriate geodesic ball.
A key element needed to guarantee the orthogonality of the trial functions Pi u1
to u1 which we will use in the sequel is the center of mass argument embodied in
Theorem 2.1 above.
Concerning the choice of trial functions Pi we proceed as follows. Thinking of
Sn as the unit sphere in Rn+1 and with the center of mass point for Ω fixed at the
north pole we take
xi
(6.2)
,
i = 1, 2, . . . , n,
Pi = g(θ)
sin θ
where θ represents the angle of a point from the positive xn+1 –axis (the direction
of the north pole). In Sn the variable θ is the
q geodesic radial variable with respect to the north pole. Division by sin θ = 1 − x2n+1 normalizes the n–vector
(x1 , x2 , . . . , xn ). The choice of g(θ), as in the Euclidean case, is determined by the
fact that we must have equality in (6.1) when Ω is a geodesic ball. For a geodesic
ball of radius θ1 , u1 = c1 y1 where y1 satisfies
(6.3)
y1 00 + (n − 1) cot θ y10 + λ1 y1 = 0
with boundary conditions y1 (0) finite and y1 (θ1 ) = 0. This is just equation (3.1)
with m = 0 and λ = λ1 . Also, u2 = c2 (xi / sin θ)y2 (for any i = 1, 2, . . . , n) where
y2 satisfies
n−1
(6.4)
y2 = 0
y200 + (n − 1) cot θ y20 + λ2 −
sin2 θ
with boundary conditions y2 (0) = y2 (θ1 ) = 0 (this is just equation (3.1) with m = 1
and λ = λ2 ). The values λ1 and λ2 are to be taken as the least eigenvalues of these
one-dimensional radial problems. By Lemma 3.1 these are the correct identifications
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THE PPW CONJECTURE IN Sn
1083
of λ1 and λ2 for our geodesic ball. One can express y1 and y2 in terms of associated
Legendre functions. In fact,
(cos θ)
y1 (θ) = (sin θ)1−n/2 Pν−(n/2−1)
1
and
(cos θ)
y2 (θ) = (sin θ)1−n/2 Pν−n/2
2
up to unimportant constant factors, where the parameters ν1 and ν2 are related to
the eigenvalues λ1 and λ2 respectively by
n(n − 2)
.
4
We follow Abramowitz and Stegun [1] in our notation here; note that their convention for associated Legendre functions makes (sin θ)1−n/2 Pν−µ , where µ = n/2 −
1 + m and m is a nonnegative integer, and not (sin θ)1−n/2 Pνµ , the “right” ndimensional generalization of the familiar associated Legendre functions Pνm , m =
0, 1, . . . , from S2 (or R3 ). This means, in particular, that when n is even Pνµ can be
substituted for Pν−µ (they are then proportional), while if n is odd Qµν can be used
(the distinction here is between µ being an integer or half an odd integer). Using
Pν−µ with µ as above circumvents this “even-odd effect”. Since we want equality
in (6.1) when Ω is a geodesic ball, using the form of u1 and u2 for a geodesic ball
we see that g(θ) must be essentially the quotient of y2 by y1 .
Let Ω ⊂ Sn be contained in a hemisphere. Let Bλ1 denote the geodesic ball
in Sn having the same value of λ1 as Ω. By Sperner’s inequality [49] (see also
[31]) λ1 (Bλ1 ) = λ1 (Ω) ≥ λ1 (Ω? ) and by the properties of λ1 for geodesic balls (in
particular, λ1 decreases as the radius of the ball increases), we see that θ1 ≤ π/2,
where θ1 denotes the geodesic radius of Bλ1 . We now set g(θ) = y2 (θ)/y1 (θ) for
0 ≤ θ ≤ θ1 , g(θ) = g(θ1 ) for θ1 ≤ θ ≤ π/2 and we extend g(θ) to a function on
[0, π] by reflection about θ = π/2. (Note that this definition makes B(θ) = g 0 (θ)2 +
(n − 1)g(θ)2 / sin2 θ a decreasing function on [θ1 , π/2], since B(θ) = (n − 1)g(θ1 )2 /
sin2 θ there.) We then apply the center of mass result (Theorem 2.1) to Ω and
G̃(θ) = g(θ)/ sin θ obtaining a choice of Cartesian coordinate axes such that
Z
xi 2
u dσ = 0,
g(θ)
for i = 1, 2, . . . , n.
sin
θ 1
Ω
ν(ν + 1) = λ +
Using the functions Pi = g(θ)xi / sin θ in (6.1) we get
Z
Z x 2
xi 2 2
i
2
2
(6.5)
g(θ)
u1 dσ ≤
(λ2 − λ1 )
∇ g(θ)
u1 dσ
sin θ
sin θ
Ω
Ω
for 1 ≤ i ≤ n, and summing on i from 1 to n we obtain
R
B(θ)u21 dσ
(6.6)
,
λ2 − λ1 ≤ R Ω
2 2
Ω g(θ) u1 dσ
where
n−1
g(θ)2
sin2 θ
as defined previously (see equation (4.2)). We observe that, in spite of the fact
that Ω is contained in a hemisphere, our use of the center of mass result may imply
that Ω does not lie in the northern hemisphere (i.e., θ would not be in (0, π/2) for
all points in Ω). To remedy this situation, we observe that since Ω is contained
(6.7)
B(θ) = g 0 (θ) +
2
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1084
MARK S. ASHBAUGH AND RAFAEL D. BENGURIA
in a hemisphere, −Ω ⊂ Sn \ Ω. Thus, if we let Ω± = {~x ∈ Ω ±xn+1 > 0}
we have Ω+ ∩ (−Ω− ) = ∅. Since g(θ) and B(θ) are both symmetric with respect
to θ = π/2 it follows that the integrals over Ω can be replaced by integrals over
Ω̃ = Ω+ ∪ (−Ω− ) with no change in their values if we agree to transplant u1 along
with Ω− to −Ω− . This follows since g and B were defined to be symmetric about
θ = π/2 and therefore they transplant into themselves. After moving the whole
problem to the northern hemisphere (where g is increasing and B is decreasing) we
can carry out all the further rearrangements in exact parallel with the Euclidean
case, encountering no further difficulties.
To conclude the proof of Theorem 1.1 we need the following two chains of inequalities. We have
Z
Z
Z
B(θ)u21 dσ =
B(θ)ũ21 dσ ≤
B(θ)? u?2
1 dσ
Ω̃
Ω
Ω?
Z
Z
(6.8)
?2
B(θ)u1 dσ ≤
B(θ)v12 dσ,
≤
Ω?
and
Z
Z
(6.9)
Ω
Z
g(θ)2 ũ21 dσ ≥
g(θ)2 u21 dσ =
Bλ1
Ω̃
Ω?
Z
≥
Ω?
g(θ)2? u?2
1 dσ
g(θ)2 u?2
1
Z
dσ ≥
g(θ)2 v12 dσ.
Bλ1
of
Here ũ1 represents u1 as transplanted to Ω̃ and v1 is the first eigenfunction
R 2
with
Dirichlet
boundary
conditions
and
normalized
so
that
u
dσ
=
−∆
on
B
λ
1
1
Ω
R
2
v
dσ.
The
functions
g
and
B
are
likewise
based
on
the
eigenfunctions
of
the
Bλ1 1
ball Bλ1 (so that the θ1 that goes into the boundary value problems (6.3) and
(6.4) that define them is the radius of the ball Bλ1 ). In each of (6.8) and (6.9),
the equality is trivial, the first inequality follows simply from rearrangement (see
Section 5 for our notation), the second inequality is by virtue of the monotonicity
properties of g(θ) and B(θ), and the last inequality follows from our Sn analog of
Chiti’s comparison result (see Section 5 for details) and also uses the monotonicity
properties of g and B again. Finally, from (6.6), (6.8), and (6.9) we obtain
R
2
Bλ B(θ)v1 dσ
= λ2 (Bλ1 ) − λ1 (Bλ1 ).
(6.10)
λ2 (Ω) − λ1 (Ω) ≤ R 1
2 2
Bλ g(θ) v1 dσ
1
Hence (since λ1 (Bλ1 ) = λ1 (Ω))
(6.11)
λ2 (Ω) ≤ λ2 (Bλ1 )
which concludes the proof of Theorem 1.1, it being clear from any of a number of
our previous inequalities that equality obtains in (6.11) if and only if Ω is itself a
ball.
Remarks. (1) Theorems 1.1 (i.e., inequality (6.11)) and 1.4 also hold under somewhat more general circumstances than for Ω contained in a hemisphere of Sn . In
particular, they continue to hold if Ω ∩ (−Ω) = ∅, or, more generally, if Ω satisfies
the “excess less than or equal to deficit property” with respect to the center of
mass as north (or south!) pole as stated in (2.13) above for all k ∈ [0, 1]. We
note, however, that even under these conditions Ω is constrained to have volume
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THE PPW CONJECTURE IN Sn
1085
no larger than that of a hemisphere. See also Remark 4 in Section 2 following the
proof of Theorem 2.1.
(2) In fact, it is enough that Ω satisfy θ1 (Bλ1 ) ≤ π/2 (or, equivalently, λ1 (Ω) ≥
n), where θ1 (Bλ1 ) is the geodesic radius of the ball Bλ1 , together with the “excess
less than or equal to deficit property” (2.13) for all k ∈ [cos θ1 (Bλ1 ), 1]. This
condition allows us to prove that λ2 (Ω) ≤ λ2 (Bλ1 ) even for certain domains which
have volume larger than that of a hemisphere (as well as covering all previous cases).
A variant of this condition also applies in the case of the Neumann problem (the
“µ1 problem”) for domains in Sn (see [9] and certain of our remarks in Section
2 above). Then θ1 (Ω? ), the geodesic radius of Ω? , should replace θ1 (Bλ1 ) in the
foregoing (and in this case we are still limited by |Ω| ≤ 12 |Sn |). The reason for these
values of θ1 is that these are the radii we end with in the respective problems, when
all is said and done.
Acknowledgements
M.S.A. is grateful for the hospitality of Thomas Hoffmann–Ostenhof and the
Erwin Schrödinger Institute (ESI) in Vienna, where some of this work was carried
out. We would also like to thank the referee for several useful remarks.
References
1. M. Abramowitz and I. A. Stegun, editors, Handbook of Mathematical Functions, National
Bureau of Standards Applied Mathematics Series, vol. 55, U.S. Government Printing Office,
Washington, D.C., 1964. MR 34:8607
2. M. A. Armstrong, Basic Topology, Springer–Verlag, New York, 1983. MR 84f:55001
3. M. S. Ashbaugh and R. D. Benguria, Log–concavity of the ground state of Schrödinger operators: a new proof of the Baumgartner–Grosse–Martin inequality, Phys. Lett. A 131 (1988),
273–276. MR 89i:81012
4. M. S. Ashbaugh and R. D. Benguria, Optimal lower bounds for eigenvalue gaps for
Schrödinger operators with symmetric single–well potentials and related results, Maximum
Principles and Eigenvalue Problems in Partial Differential Equations, P. W. Schaefer, editor,
Pitman Research Notes in Mathematics Series, vol. 175, Longman Scientific and Technical,
Harlow, Essex, United Kingdom, 1988, pp. 134–145. MR 90c:35157
5. M. S. Ashbaugh and R. D. Benguria, Proof of the Payne–Pólya–Weinberger conjecture, Bull.
Amer. Math. Soc. 25 (1991), 19–29. MR 91m:35173
6. M. S. Ashbaugh and R. D. Benguria, A sharp bound for the ratio of the first two eigenvalues
of Dirichlet Laplacians and extensions, Annals of Math. 135 (1992), 601–628. MR 93d:35105
7. M. S. Ashbaugh and R. D. Benguria, A second proof of the Payne–Pólya–Weinberger conjecture, Commun. Math. Phys. 147 (1992), 181–190. MR 93k:33002
8. M. S. Ashbaugh and R. D. Benguria, Isoperimetric inequalities for eigenvalue ratios, Partial
Differential Equations of Elliptic Type, Cortona, 1992, A. Alvino, E. Fabes, and G. Talenti,
editors, Symposia Mathematica, vol. 35, Cambridge University Press, Cambridge, 1994, pp.
1–36. MR 95h:35158
9. M. S. Ashbaugh and R. D. Benguria, Sharp upper bound to the first nonzero Neumann
eigenvalue for bounded domains in spaces of constant curvature, J. London Math. Soc. (2)
52 (1995), 402–416. MR 97d:35160
10. M. S. Ashbaugh and R. D. Benguria, On the Payne–Pólya–Weinberger conjecture on the ndimensional sphere, General Inequalities 7 (Oberwolfach, 1995), C. Bandle, W. N. Everitt,
L. Losonczi, and W. Walter, editors, International Series of Numerical Mathematics, vol. 123,
Birkhäuser, Basel, 1997, pp. 111-128. MR 98k:35139
11. M. S. Ashbaugh and H. A. Levine, Inequalities for the Dirichlet and Neumann eigenvalues
of the Laplacian for domains on spheres, Journées “Équations aux Dérivées Partielles”
(Saint–Jean–de–Monts, 1997), Exp. No. 1, 15 pp., École Polytechnique, Palaiseau, 1997.
12. F. E. Baginski, Ordering the zeroes of Legendre functions Pνm (z0 ) when considered as a
function of ν, J. Math. Anal. Appl. 147 (1990), 296–308. MR 91k:33006
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
1086
MARK S. ASHBAUGH AND RAFAEL D. BENGURIA
13. F. E. Baginski, Comparison theorems for the ν-zeroes of Legendre functions Pνm (z0 ) when
−1 < z0 < 1, Proc. Amer. Math. Soc. 111 (1991), 395-402. MR 91i:33003
14. C. Bandle, Isoperimetric Inequalities and Applications, Pitman Monographs and Studies in
Mathematics, vol. 7, Pitman, Boston, 1980. MR 81e:35095
15. C. Bandle and M. Flucher, Table of inequalities in elliptic boundary value problems, Recent
Progress in Inequalities (Niš, 1996), G. V. Milovanovic, editor, Mathematics and its Applications, vol. 430, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998, pp. 97–125.
MR 99h:35028
16. B. Baumgartner, Level comparison theorems, Annals of Physics 168 (1986), 484–526.
MR 87j:81046
17. B. Baumgartner, Relative concavity of ground state energies as functions of a coupling
constant, Phys. Lett. A 170 (1992), 1–4. MR 93i:81022
18. B. Baumgartner, H. Grosse, and A. Martin, The Laplacian of the potential and the order of
energy levels, Phys. Lett. B 146 (1984), 363–366. MR 85k:81158
19. B. Baumgartner, H. Grosse, and A. Martin, Order of levels in potential models, Nucl. Phys.
B 254 (1985), 528–542. MR 87a:81024
20. G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, fourth edition, Wiley, New
York, 1989. MR 90h:34001
21. Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Grundlehren der mathematischen
Wissenschaften 285, Springer–Verlag, Berlin, 1988. MR 89b:52020
22. I. Chavel, Lowest-eigenvalue inequalities, Proc. Symp. Pure Math., vol. 36, Geometry of
the Laplace Operator, R. Osserman and A. Weinstein, editors, Amer. Math. Soc., Providence,
Rhode Island, 1980, pp. 79–89. MR 81f:58039
23. I. Chavel,
Eigenvalues in Riemannian Geometry, Academic Press, New York, 1984.
MR 86g:58140
24. G. Chiti, Orlicz norms of the solutions of a class of elliptic equations, Boll. Un. Mat. Ital.
(5) 16–A (1979), 178-185. MR 80h:35004
25. G. Chiti, A reverse Hölder inequality for the eigenfunctions of linear second order elliptic
operators, J. Appl. Math. and Phys. (ZAMP) 33 (1982), 143–148. MR 83i:35141
26. G. Chiti, An isoperimetric inequality for the eigenfunctions of linear second order elliptic
operators, Boll. Un. Mat. Ital. (6) 1–A (1982), 145–151. MR 83i:35140
27. G. Chiti, A bound for the ratio of the first two eigenvalues of a membrane, SIAM J. Math.
Anal. 14 (1983), 1163–1167. MR 85f:35158
28. R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. I, Interscience Publishers,
New York, 1953. MR 16:426a
29. J. Dugundji, Topology, Allyn and Bacon, Boston, 1966. MR 33:1824
30. G. Faber, Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher
Spannung die kreisförmige den tiefsten Grundton gibt, Sitzungsberichte der mathematischphysikalischen Klasse der Bayerischen Akademie der Wissenschaften zu München Jahrgang,
1923, pp. 169–172.
31. S. Friedland and W. K. Hayman, Eigenvalue inequalities for the Dirichlet problem on spheres
and the growth of subharmonic functions, Comment. Math. Helvetici 51 (1976), 133–161.
MR 54:568
32. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, second edition, Cambridge University Press, Cambridge, 1952. MR 13:727a
33. E. M. Harrell II and P. L. Michel, Commutator bounds for eigenvalues, with applications to
spectral geometry, Commun. Partial Diff. Eqs. 19 (1994), 2037–2055. MR 95i:58182
34. P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory, With Applications to
Schrödinger Operators, Applied Mathematical Sciences, vol. 113, Springer–Verlag, New York,
1996. MR 98h:47003
35. J. G. Hocking and G. S. Young, Topology, Addison–Wesley, Reading, Massachusetts, 1961.
MR 23:A2857
36. T. Kato, Perturbation Theory for Linear Operators, 2nd edition, Grundlehren der mathematischen Wissenschaften 132, Springer–Verlag, Berlin, 1976. MR 53:11389
37. E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann.
94 (1925), 97–100.
38. E. Krahn, Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen, Acta Comm.
Univ. Tartu (Dorpat) A9 (1926), 1–44. [English translation: Minimal properties of the sphere
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
THE PPW CONJECTURE IN Sn
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
1087
in three and more dimensions, Edgar Krahn 1894–1961: A Centenary Volume, Ü. Lumiste
and J. Peetre, editors, IOS Press, Amsterdam, The Netherlands, 1994, pp. 139–174.]
J. R. Munkres, Topology, A First Course, Prentice–Hall, Englewood Cliffs, New Jersey, 1975.
MR 57:4063
J. R. Munkres, Elements of Algebraic Topology, Addison–Wesley, Menlo Park, California,
1984. MR 85m:55001
R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), 1182–1238.
MR 58:18161
L. E. Payne, G. Pólya, and H. F. Weinberger, Sur le quotient de deux fréquences propres
consécutives, Comptes Rendus Acad. Sci. Paris 241 (1955), 917–919. MR 17:372d
L. E. Payne, G. Pólya, and H. F. Weinberger, On the ratio of consecutive eigenvalues, J.
Math. and Phys. 35 (1956), 289–298. MR 18:905c
J. W. S. Rayleigh, The Theory of Sound, second edition revised and enlarged (in two volumes),
Dover Publications, New York, 1945 (republication of the 1894/1896 edition). MR 7:500e
M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. IV: Analysis of
Operators, Academic Press, New York, 1978. MR 58:12429c
R. D. Richtmyer, Principles of Advanced Mathematical Physics, vol. II, Springer–Verlag,
New York, 1981. MR 84h:00024b
K. T. Smith, Primer of Modern Analysis, Bogden and Quigley, Tarrytown–on–Hudson, New
York, 1971.
E. H. Spanier, Algebraic Topology, McGraw–Hill, New York, 1966. MR 35:1007
E. Sperner, Zur Symmetrisierung von Funktionen auf Sphären, Math. Z. 134 (1973), 317–
327. MR 49:5310
G. Szegő, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech.
Anal. 3 (1954), 343–356. MR 15:877c
G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa (4) 3 (1976),
697–718. MR 58:29170
H. F. Weinberger, An isoperimetric inequality for the n-dimensional free membrane problem,
J. Rational Mech. Anal. 5 (1956), 633–636. MR 18:63c
E. F. Whittlesey, Fixed points and antipodal points, Amer. Math. Monthly 70 (1963), 807–821.
MR 28:4532
Department of Mathematics, University of Missouri, Columbia, Missouri 65211–0001
E-mail address: [email protected]
Departamento de Fı́sica, P. Universidad Católica de Chile, Casilla 306, Santiago 22,
Chile
E-mail address: [email protected]
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